FACOLTÀ DI INGEGNERIA
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE
Corso di Dottorato in
Automatica, Robotica e Bioingegneria
Scuola di Dottorato in Ingegneria
"Leonardo da Vinci"
SSD: ING-INF/04
Ph.D. Dissertation on
Soft Robotics: from Optimality Principles
to Technology Readiness
Candidate Student:
Manolo Garabini
Supervisors:
Prof. Antonio Bicchi
For many years robotic researchers have been focused on developing robot sensing and control, while the actuation was left at the level of position controlled servomotors. In the last two decades this trend is changed, and actuators gained a richer dynamical behaviour. From Series Elastic Actuation (SEA) in which an elastic element was in-terposed between the motor and the link, to Variable Stiffness (VSA) and variable damping actuators in which the actuation unit can phys-ically change its dynamical parameters, i.e. stiffness and damping. This new actuation paradigm, called Soft Robotics, is perceived as an enabling factor to build robots that are robust, efficient, and have high peak performance. Robot that should be able to perform everyday-life tasks and to safely coexist with humans.
In such a background this work first concerns with: i) to com-pare different actuation paradigms to effectively evaluate the poten-tial benefit of soft actuation w.r.t. conventional rigid motors; ii) to understand how to manage the additional design and control degrees of freedom of soft robots.
Optimal Control (OC) theory has been chosen as the fundamental tool to accomplish the task. This choice is motivated by two main reasons: i) on one side OC provides an absolute performance refer-ence that factorises the control design out of the equation, hrefer-ence it furnish a principled basis to compare the performance of different
careful analysis of results, obtained through either analytic or numer-ical techniques, allows to distillate laws summarising control policies that can be applied to classes of tasks.
In this thesis, the methodology described above has been applied to show that there exists an optimal linear spring that maximises the peak speed, the energy efficiency, or the force tracking error under ro-bustness constraint of a SEA actuator and its stiffness value depends on motor constraints (e.g. speed and torque), task parameters (e.g. terminal time), joint trajectories, environment parameters (e.g. the stiffness of the environment). This fact is per se a strong motivation to employ VSA to obtain maximum performances for different kind tasks and environment. Moreover it has been possible to derive a complete analytical solution for the optimal control problem of po-sition and stiffness control of one DoF robot. Through the analysis of these results the stiffness optimal control policy (dependent on the link motion) can be summarised by the law: stiff when speed-up, soft when slow-down. Experimental tests showed: i)a more than doubled peak speed of the SEA (also tested on a two DoFs series elastic ma-nipulator) w.r.t the rigid motor, and a further 30% improvement with the VSA, ii) a good agreement between theoretical and experimental cost functionals for the study on the energy efficiency.
Once suitable control laws and field of application for soft Robotics has been defined it is necessary to bring this new actuation paradigm to the proper technology readiness level to be accessible to the whole robotic community not only to researchers focused on the robot design aspect.
In this work this problem has been tackled via: i)the definition of the main functional specifications of a variable impedance actuator that have been collected in a datasheet; ii) the development of a modular open-source and low-cost variable stiffness robotic platform: the qb move.
accessible to a large number of researchers. The qb move (today it is a product commercialised by qb robotics s.r.l.) is intended to be used as a rapid prototyping platform for experimental tests, and it allows to drastically reduce the time to experiment to test new soft robotics applications.
Many people substantially contributed to the work presented in this thesis: it has been a pleasure to walk through last three years with all of them.
It has been a pleasure to spend night and days with Manuel, Gior-gio, Lorenzo, Andrea, Fabio, Laura, Felipe, Gian Maria, Alberto, Mat-teo, Alessandro and Paolo. It has been a pleasure to learn with them how new ideas and passion can produce real things, to share unfor-gettable moments, and to be enriched by their friendship.
It has been a pleasure to collaborate with students like Andrea, Michele, Leonardo, Alessio, Riccardo, Leonardo, Davide, and Enrico that have brought new energies and freshness in the laboratory.
It has been a pleasure to learn from Fabio Vivaldi a priceless source of experience.
It has been a pleasure to know people from all over the world like Carlos, Arash, Alexandra, Manuel, Kamilo, Yongtae, Lisha, and Navvab.
It has been a pleasure to be advised by bright researchers and professors like Tsagarakis, Pallottino, Gabiccini, Fagiolini and Greco.
Thanks to everyone.
Thanks, especially, to Antonio Bicchi because he gave me and gives me the chance to do what I like.
Abstract v
Acknowledgement ix
Introduction 1
I
Soft Robotics: Optimality Principles
17
1 Optimal Control For Maximising Peak Speed 19
1.1 O. Principles in Stiffness Control: The VSA Hammer . 19
1.1.1 Problem Definition . . . 21
1.1.2 Optimal Control: SEA . . . 23
1.1.3 Optimal Control: VSA . . . 26
1.1.4 Experimental Results . . . 34
1.2 O. Principles in Stiffness Control: The VSA Kick . . . 37
1.2.1 Problem Definition . . . 39
1.2.2 Optimal Control: SEA . . . 41
1.2.3 Optimal Control: VSA . . . 46
1.2.4 Experiments . . . 52
1.3 O. C. for Maximising Velocity of the CompAct C. A. . 56
1.3.2 Control Strategy . . . 61
1.3.3 Simulation Results . . . 70
1.4 O. Principles in Design and Control of Jumping Robots 76 1.4.1 Dynamic Models . . . 79
1.4.2 Problem Definition . . . 83
1.4.3 Problem Solution . . . 87
1.4.4 Simulation Results . . . 96
1.5 O.C. of a Planar SEA Manipulator . . . 103
1.5.1 Problem Definition . . . 104
1.5.2 Problem Solution . . . 106
1.5.3 Examples . . . 110
1.5.4 Simulation Results . . . 115
1.5.5 Experimental Tests . . . 126
2 Energy Efficiency Maximisation in Cyclic Tasks 135 2.1 Background . . . 139
2.1.1 Fully Actuated and Underactuated Systems . . 139
2.1.2 Performance Indices . . . 141
2.1.3 Problem Definition . . . 142
2.2 Optimisation of Stiffness and Preload . . . 142
2.2.1 Stiffness optimisation for SEA . . . 143
2.3 One-link Robot . . . 147
2.4 Two-link Robot: Simulation Results . . . 150
2.5 Experimental Results . . . 154
2.5.1 Test Protocol and Procedure . . . 154
2.5.2 Results . . . 157
2.5.3 Model–Free Application . . . 167
3 Optimal Force Tracking s.t. Robustness Constraints 171 3.1 Problem Formulation . . . 174
3.2 A Remark . . . 178
3.3 Methodology . . . 182
3.4 Open Loop Optimal Control . . . 185
II
Soft Robotics: Technology Readiness
191
4 Functional Specifications of a VSA 193
4.1 Key Features of VSA Performance . . . 194
4.2 Controlling VSAs . . . 198
4.3 Other Characteristics of VSAs . . . 203
4.4 A VSA Datasheet . . . 204
4.5 Examples . . . 209
4.5.1 Example 1: a Multi-Material Cutting Tool-Head.209 4.5.2 Example 2: a Multipurpose Tool-Head. . . 213
4.6 Guidelines to Characterise a VSA . . . 216
4.6.1 Quasi-Static Load Cycles . . . 216
4.6.2 Step Command . . . 221
5 VSA -CUBE: Design of a Servo VSA 225 5.1 The VSA-CUBE . . . 227
5.2 Mechanical Design . . . 230
5.3 Electronics and Control Interface . . . 232
5.4 Mathematical Model . . . 233
5.5 The CUBE-BOT a Multi-DoF Platform . . . 236
5.6 Experiments . . . 237
5.7 From VSA-Cube to Biologic Muscles . . . 244
5.7.1 Muscle Force Model . . . 245
5.7.2 Agonistic-Antagonistic VSA Model . . . 247
6 Design of a Variable Impedance Actuator 251 6.1 Variable Impedance Actuator Layouts . . . 252
6.2 Variable Damper Design . . . 255
6.2.1 Requirements . . . 255
6.2.2 Mechanical Design . . . 258
6.3 Variable Damping Experimental Tests . . . 263
A VIA@play:Drumming 269
A.1 Drum Stroke . . . 272
A.1.1 Single Stroke and Double Stroke . . . 272
A.1.2 Drum Roll . . . 272
A.1.3 Simulation . . . 275
A.2 Variable Stiffness Actuators . . . 277
A.3 Finding Rolling Stiffness for Double Stroke . . . 277
A.4 Experimental Setting . . . 279
A.5 Experimental Results . . . 280
A.6 Conclusion and Future Works . . . 282
B VIA@work:Drawing on a Wavy Surface 285 B.1 Problem Definition . . . 287
B.2 Optimal Workspace Stiffness . . . 289
B.3 Optimal Joint Stiffness . . . 294
B.4 Experimental Tests . . . 295
C VIA@work:Towards Variable Impedance Assembly 303 C.1 Problem Definition . . . 307
C.1.1 Kinematics . . . 307
C.1.2 Dynamics . . . 308
C.1.3 The Task . . . 308
C.2 Problem Solution . . . 309
C.2.1 From Parallel to Serial Manipulation . . . 309
C.2.2 Search Algorithm . . . 310
C.2.3 Insertion . . . 311
C.2.4 Control . . . 314
C.3 Simulation and Experimental Results . . . 316
Up to now, robots that accomplish real-world applications are con-fined to operate in rigidly structured environments, without the pres-ence of human beings. The scenario where robots perform complex tasks at a level comparable to the human one is still left to fiction movies.
The author is aware that this challenge will face robotic researchers for some decades in the next future. Still he hopes that this work can constitute a small, yet tangible advancement to develop robots that help humans in everyday life.
Effective motivations are clarified below through examples of tasks showing how far ahead humans are w.r.t. state-of-the-art robots, how high the impact of having robots approaching human performance could be and how much the research community feels the need to close the gap. Such examples will be used to derive the features that play a key role in the successful execution of such tasks by humans (namely robustness, efficiency, high peak performances), features that need to be embodied in the next generation of robots.
As a matter of fact there is not a robot that could be useful in the disaster scenarios unfortunately happened in recent years, an em-blematic example was the Fukushima case, but we also could think to uncountable fires, earthquakes, tsunami happened in last years. Nowadays human beings are the only option for operations in these
catastrophic scenarios. In Fig. 1 some of the tasks that are commonly executed in such situations are shown.
Figure 1: The collection of pictures shows (from the top left to right bottom) firemen and people of the Civil Protection at work: hitting the top of a roof with an axe, hitting a wall with a hammer, un-loading heavy objects, using a water pump, carrying objects while climbing stairs, removing an obstacle, climbing on a challenging ter-rain, rescuing people on the snow, opening passages removing debris of a collapsed building. In the execution of these tasks humans ex-ploit some intrinsic features of their body, such as power, robustness, explosive energy, efficiency, etcetera
The robotic research community does not underestimate the im-portance of the challenge of realising a robot able to help or substitute human operators that accomplish such dangerous tasks when these events happens. One of the most important initiatives born from this need is the DARPA Robotics Challenge [W21]: a worldwide compe-tition of which the goal is:
dangerous, degraded, human-engineered environments. The program will focus on robots that can use available human tools, ranging from hand tools to vehicles. The program aims to advance the key robotic technologies of supervised autonomy, mounted mobility, dismounted mobility, dexterity, strength, and platform endurance. Supervised au-tonomy will be developed to allow robot control by non-expert oper-ators, to lower operator workload, and to allow effective operation despite low fidelity (low bandwidth, high latency, unreliable) commu-nications. [W21]
And it is expected that
[...] This technology will improve the performance of robots that oper-ate in the rough terrain and austere conditions characteristic of dis-asters, and use vehicles and tools commonly available in populated areas. [W21].
Some of the tasks composing the DRC are shown in Fig. 2: operate a valve and use a jackhammer [W21]. Other tasks include walking over rough terrain, walking on an unstable bridge, use of tools (drills, screwdriver, etc.), drive a car, open or close doors, open a gap in walls,etcetera. The differences in performance that exist between a human and a state-of-the-art robot (preventing robots to be useful in such environmental conditions) are not only at the cognitive level, one of the most exciting challenges in robotic research, but also at the intrinsic mechanical level. The capabilities that a biologic system can show, such as running, jumping, manipulating, and interacting with the surrounding world, hitting, etcetera are very far beyond what a robot can do. To reduce this gap it is essential that some features of the state-of-the-art robots are substantially improved: i) the level of robustness (both at intelligence and structural level), ii) the energy efficiency, iii) the flexibility to cope with unstructured environment,
Figure 2: The picture, extracted from the official website of the DARPA Robotics Challenge [W21], shows two of the main tasks that robots participating the competition have to solve: operate a valve and use a jackhammer.
and iv) the peak performances, e.g. in terms of speed and torque. These barriers can be overcome by improving three of the many factors that contribute to the realisation of an effective robotic system: sensing (from the sensors to perception algorithms), control (from the lowest level to artificial intelligence), and actuation system.
For many years robotic researchers have been focused on develop-ing robot sensdevelop-ing and control, while the actuation was left at the level of position or torque controlled servomotors. In the last three decades this trend is changed, and actuators gained a richer behaviour.
The first step was achieved by conferring the capability to the robot of controlling the impedance at the end effector via the Ac-tive Impedance Control approach [46]. The novelty of this technique is concentrated in the control of the actuation system but it leaves almost unchanged the mechanics of the prime movers.
However undoubtable the advantages of Active Impedance con-trol (technologically available today) w.r.t. position servo-actuators are, still it presents substantial limitations: low structural robustness against unpredictable impacts and high bandwidth required to the
The intrinsic dynamic of existing actuators have been enriched to overcome such limitations via the introduction of additional compo-nents.
In Series Elastic Actuation (presented in [47]) a elastic element (with constant stiffness) was interposed between the motor and the link, while in Variable Stiffness (for a complete and recent review see [A1]) and Variable Damping (see [48]) actuators the actuation unit can physically change its dynamical parameters, i.e. stiffness and damping.
This new actuation paradigm, called Soft Robotics, is becoming more and more popular among robotic researchers and now it is per-ceived as the new technological wave for building a robot generation aiming to take a substantial step in filling the gap with humans.
Fig.s 3, 4, 5 and 6 show some of the most recent advancements in Robotics development. All these images show robots based and developed following the ideas behind the Soft Robotics, as for example the introduction of passive elastic elements in the architecture of the robots (Fig.s 4, 5 and 6), or actively controlling the impedance of the system (Fig. 3).
Figure 3: The sequence shows the new ASIMO [W22] running, with a velocity of [9km/h], and jumping [W23]. In this case each joint is controlled with an active impedance controller, an intrinsic elasticity is provided from the Harmonic Drive gearboxes [W24].
Figure 4: The sequence shows the robustness of the equilibrium con-trol and stabilisation concon-trol of the robot COMAN [W25], developed at Italian Institute of Technologies (IIT) [W26]. Each joint has a pas-sive elastic element in series with the output shaft of the actuator. A sophisticated impedance control is employed to have these kinds of performance.
Figure 5: The sequence shows the robot PET-MAN (Boston Dynam-ics [W27]) performing some advanced tasks such as: climb an obstacle or walk in uneven terrain [W28]. The robot have a pneumatic actua-tion system, for this reason has an intrinsic passive compliance.
Today the Series Elastic concept is available for research labora-tory. An example is Baxter (see Fig. 7) a humanoid torso realised by RethinkRobotics [W31] equipped with two 7-DoF series elastic arms. On the other hand most recent ideas like Variable Impedance actuation are still at the test phase of prototypes in lab environment.
the new DLR Hand-Arm system [W30]. In this device each joint is equipped with a Variable Stiffness Actuator, also the hand has a struc-ture where each joint has a passive compliance system of actuation.
Figure 7: Baxter: the adaptive, collaborative manufacturing robot realised by RethinkRobotics
Soft Robotics discloses a completely new range of potentialities at the cost of adding complexity, from both the fabrication and the control point of views. This challenges the robotic community to better understand when and how to use soft actuation and to make Soft Robotics usable by researchers and practitioners out the field of robot design.
Given this background, the objective of this thesis are:
• to compare different actuation paradigms to effectively evaluate the potential benefit of soft actuation w.r.t. conventional rigid motors.
• to understand how to manage the additional design and control degrees of freedom of soft robots.
• to provide a low-cost ready-to-use soft robotic platform to let the interested users explore on their own the potential of this novel actuation approach
Optimal Control (OC) theory has been chosen as the fundamental tool to reach the first two objectives. This choice is motivated by two main reasons:
• on one side OC provides an absolute performance reference that factorises the control design out of the equation, hence it fur-nish a principled basis to compare the performance of different system designs;
• on the other side, OC is a key element in understanding plan-ning and control methodologies for soft actuators. A careful analysis of results, obtained through either analytic or numer-ical techniques, allows to distillate laws summarising control policies that can be applied to classes of tasks.
This approach is, in principle, similar to the one presented in the work [49] by Bicchi and Tonietti where the problem of performing a rest-to-rest motion in minimum time under safety constraint, named
ation and soft actuation have been compared. It has been shown that there exists an optimal spring (with constant stiffness) that minimises the time to accomplish the task, and second that the performance can be improved if the stiffness is changed during the task and the optimal control policy for the stiffness can be summarised in the rule: fast and soft, stiff and slow (see Fig. 8).
Figure 8: The picture shows the optimal velocity (blu line) and the stiffness (purple line) profiles for the safe brachistochrone problem
Once (some of the) fields of application where soft actuators can be profitably exploited has been explored, it is necessary make soft robotics accessible to researchers also not focused on the robot design aspect. This point has been tackled:
• by defining the main features of a variable impedance actuator and by organising them in a data-sheet
• by developing of a modular, open-source, and low-cost variable stiffness robotic platform: the qb move.
The classification and the data-sheet allows a not-in-the-field but skilled user to have, at a glance, the parameters and the information to evaluate if and what soft actuator is useful to its scopes and how to integrate it in his project. E.g. for the Variable Stiffness Actuators this work ended up with a data-sheet in which the operational space of the device is no more defined by just speed and torque (as usual for a conventional electric motor) but by a 3-D volume described by speed, torque, and stiffness. Moreover, use cases exempla (multi-material cutter, multi-purpose tool head) are carried out to show how it is possible to translate task requirements in specifications for this new class of actuation.
The qb move is intended to be used as a rapid prototyping plat-form for experimental tests, and it allows to drastically reduce the required resources to get access to soft robotics technology. Experi-mental validations and mathematical models of the system employed in single (hammering, cutting, drumming) and multi degrees of free-dom tasks (bimanual assembly and drawing on an uneven surface), are reported.
According to the definitions of the Technology Readiness Levels (TRL) of the U.S. Department of Defence [W32], implemented also by the European Union within the research and innovation programme Horizon 2020 [W33], the newest Soft Robotics concepts, e.g. state-of-the-art Variable Stiffness Actuation, could be set at TRL 4: technol-ogy validated in laboratory. Several prototypes of Variable Stiffness Actuators (see [A1] for an updated review) have been built, and their functionality has been validated in laboratory environment.
During this thesis a big effort has been put to make the Vari-able Stiffness technology availVari-able to interested practitioners. This regarded not only the development of a novel device (the VSA-Cube) to bring it to be a product (the qb move), but also the realisation of a community, the Natural Motion Initiative [W34] that collects useful information, project, software, drawings about Soft Robotics.
During this work several demonstrations of working robots built with the qb move platform has been given outside the laboratory
en-TechFest 2014, Mumbai, India; ERF 2014, Rovereto, Italy).
These facts let the author think that this work has brought the Variable Stiffness Actuation to TRL 6: technology demonstrated in relevant environment.
Implication of this work are:
• Optimality principles in design and control of soft robots are making it clear how and what actuation paradigms should be used in which tasks. Examples are maximising the actuator peak speed, maximising energy efficiency, obtain robust and flexible actuators. These results makes robotic community in-terest in soft robotics grows year after year.
• Soft Actuator classification and the datasheet definition have substantially contributed to unify and standardise the language around this topic making this technology and literature acces-sible to a large number of researchers.
• The qb move platform has been tested in several experimental validations with single DoF and multi DoF robot designs. It is open-source and low-cost and it drastically reduces the time to experiment. Today it is a product commercialised by qb robotics s.r.l. [W35], a start-up company of which the author is one of the founders.
• Since the mechanical realisation of the qb move is very similar to the antagonist arrangement of biological muscular groups, it turns out that the control approach of the qb move (reference position and stiffness preset) resembles a robotic declination of the Equilibrium Point Hypothesis (one of the most popular hy-potheses in the motor control literature). This possibly leaves room for using the qb move platform to better understand
hu-man motor control mechanisms and to test their applicability in the robotic field.
Main Contributions and their correlation with international research projects The main contributions of this thesis are:
• Formalisation of optimisation and optimal control problems for maximising the peak speed (Chapter 1), the energy efficiency (Chapter 2), and the force tracking error (Chapter 3) for Soft Robots. This is a result produced inside the European Commu-nity’s projects VIACTORS [P42], SAPHARI [P43], and WAL-KMAN [P44].
• There exists an optimal linear spring that maximises the peak speed (Chapter 1), the energy efficiency (Chapter 2), and the force tracking error under robustness constraint (Chapter 3) of a SEA actuator and its stiffness value depends on motor con-straints (e.g. speed and torque), task parameters (e.g. terminal time), joint trajectories, environment parameters (e.g. the stiff-ness of the environment). This fact is per se a strong motivation to employ Variable Stiffness Actuators to obtain maximum per-formances for different tasks and environment.
• Complete analytical solution for the optimal control problem of maximising the peak speed for position and stiffness control of one DoF Variable Stiffness robot (Section 1.1). Through the analysis of these results the stiffness optimal control policy (de-pendent on the link motion) can be summarised by the law: stiff when speed-up, soft when slow-down.
• Theoretical results obtained through analytical formulas and simulations has been experimentally tested obtaining: i)a more than doubled peak speed of the SEA w.r.t the rigid motor, and a further 30% improvement with the VSA (Section 1.1 and 1.2), ii) a good agreement between theoretical and experimental
2). Moreover experimental tests with current measurements has been conducted and they confirmed that the predicted level of efficiency reflects the real consumption of the robot.
• Translation of the optimal control problem of maximising the peak speed of a Variable Damping actuator with control bounds in a series of convex optimisation problem. This allows to over-come both rigid and Series Elastic actuation performance for a variety of tasks with different terminal times by appropriately managing the releasing of elastic energy (Section 1.3).
• Translation of the optimal control problem of maximising peak speed with linear actuator dynamics (linear SEA systems or lin-ear PEA systems) and convex state and control path constraints (e.g. the linear torque-speed characteristic of the motor) in a convex optimisation problem. For example this allows to com-pare series elastic with parallel elastic design in the jumping task (Section 1.4).
• Translation of a class of nonlinear optimal control problems into an optimisation problem that involves switching times. This, employed in an on-line phase plus a off-line phase, allows to efficiently and reliably solve the optimisation problem. The al-gorithm has been successfully tested in a nonlinear model pre-dictive control framework in simulation and experimental tests with a two DoF SEA arm (Section 1.5).
• Provided that the stiffness and inertia actuation matrices are di-agonal, analytical solutions for spring stiffness and pre-load (in both cases of SEA and PEA actuation) that minimise squared torque or squared power cost functionals are given as function of the joint trajectories. This result allows to recast the general problem involving both joint trajectories and actuation param-eters to a simpler one involving joint trajectories only (Chapter 2).
• Introduction of a possible standard data-sheet for a specific ty-pology of robotic muscles, variable stiffness actuators. It is a for-malisation where the main functional parameters and the main characteristics are summarised and arranged in an organised and practical way, taking in to account, as first objective, the user’s point of view (Chapter 4). The work done in this part of the thesis is a result produced inside the European Commu-nity’s projects VIACTORS [P42] and SAPHARI [P43], with the collaboration of all its partners.
• Introduction of a new affordable and simple variable stiffness actuator, VSA-Cube. The actuator is developed to be the ba-sic component of a robotic platform, VSA-CubeBot, developed for a diffusion of variable stiffness technology. The concept of variable stiffness servo actuator is introduced (Chapter 5). The VSA-CubeBot represents, also, one of the first multi degrees of freedom platform equipped with Variable Stiffness Actuators. It is a result produced inside the European Community’s projects VIACTORS [P42] and SAPHARI [P43].
• Introduction of one of the first variable impedance actuator. This is a system where both stiffness and damping can be changed simultaneously and independently (Chapter 6). It is a result produced inside the European Community’s project SAPHARI [P43].
• Formalisation and implementation of experiments multi degrees of freedom with a robotic platform, VSA-CubeBot, totally equipped with variable stiffness actuators (Chapters C and B). It is a re-sult produced inside the European Community’s projects VI-ACTORS [P42] and SAPHARI [P43].
Structure of the thesis The thesis is organised as follow:
Part one deals with optimisation and optimal control control prob-lems for maximising peak speed 1, energy efficiency 2 and robustness
the peak speed at a given terminal position and free/given terminal time is solved for a simple dynamical model of Series Elastic and Variable Stiffness actuator. In section 1.3 the problem for fixed ter-minal time is translated into a series of convex optimisation problems for a Variable Damping actuator. In section 1.4 the peak speed op-timisation problem is treated for Series Elastic and Parallel Elastic configurations tacking into consideration also state and control path constraints (e.g. contact constraints, torque-speed motor character-istic, etc.). In section 1.5 the same problem is treated for a 2 DoFs robotic arm with Series Elastic actuators. The optimal control prob-lem (for a class of systems) is translated in an optimisation probprob-lem in which the optimisation variables are the switching times. Experi-mental tests are carried out exploiting the nonlinear model predictive control approach.
Chapter 2 provides, under suitable hypothesis on the structural properties of the system, analytical solutions for spring stiffness and pre-load (in both cases of SEA and PEA actuation) that minimise squared torque or squared power cost functionals in terms of the joint trajectories. Moreover an experimental evaluation (with current mea-surements) of theoretical results confirms predicted trend.
In Chapter 3 The role of compliant actuators in force control tasks has been explored showing that compliant actuators behave better than rigid ones in rejecting disturbances and reducing the transmis-sion force. The existence of an optimal stiffness value depending on the environment characteristics strongly motivates the use of Vari-able Impedance Actuators (VIA) to obtain optimised performance in different environments.
Part two deals with: the classification of existing working princi-ples and designs of soft robots and the definition of the main functional specifications of a variable impedance actuator; and the development of a modular open-source and low-cost variable stiffness robotic plat-form: the qb move. In chapter 4 there is an explanation of the main mechanical and control parameters constituting a robotic muscle, with
particular attention to one possible standardisation of the functional specifications an their application in use cases from the user’s point of view.
Chapters 5 and 6 deal with the design of a small, economic and affordable robotic platform, useful for the diffusion and test of soft robotics muscles. This approach culminated in the realisation of the first variable impedance actuator for robots and in the qb move the patented VSA (EP2444207 [A2]) commercialised by qb robotics s.r.l. that has been derived by the actuator described in chapter 5.
Applications in which the qb move is used in single and multi DoFs robot are shown in the chapters A, C, and B.
Clarification This thesis collects the majority of the material (pub-lished, in press or under submission) that the author has contributed to produce during his Ph.D..
This work touched several aspects: from the mathematical syn-thesis of models and control algorithms, to the electromechanical re-alisation of complex robotic systems and experimental tests.
The author’s work has been focused on one side on the modelling, optimisation and optimal control theoretical and experimental work, on the other on the ideation and the mechanical design of variable stiffness and impedance actuators.
In particular the author definitely highlights that without a fruitful collaboration with a team of valuable people much of this work (e.g. software and firmware development, electronic design, fabrication, low level control) and ideas (often outcomes of long discussions) would not be realised.
Soft Robotics: Optimality
Principles in Design and
Chapter
1
Optimal Control For Maximising
Peak Speed
1.1
Optimality Principles in Stiffness
Con-trol: The VSA Hammer
The text of this section is adapted from:
[A3] Garabini M, Passaglia A, Belo F, Salaris P, Bicchi A. Opti-mality principles in variable stiffness control: The VSA hammer. In: International Conference on Intelligent Robots and Systems (IROS), 2011.
Variable stiffness Actuators (VSA) have been designed to over-come the limits of conventionally actuated robots in terms of safety [49], e.g., in human robot interaction, and for operating in an unstruc-tured environment. After the MIA [50], a device able to slowly adjust its output shaft stiffness, the first prototype able to change stiffness quickly enough to use this capability during a task was the VSA-1 [51]. More recently, actuators with better performances have been
devel-oped: the VSA-HD [52], the QA-Joint [53] and the AwAS [54]. Dur-ing the last years the research community has recognised the impor-tance of compliant actuators in the context of optimisation problems. First, the optimal control problem of performing a rest to rest posi-tion task under a safety constraint in minimum time is solved in [49] (namely the safe brachistochrone problem). This solution shows that the VSAs achieve better performance with respect to conventional actuators and to Series Elastic Actuators (SEAs). This improvement is significant when the link and end effector inertia are small, as de-scribed in [55] and [56]. However, the true potential of performance improvement embedded in variable stiffness actuation is still to be explored, as suggested by examples in nature. The capability of soft actuators (e.g., SEAs) to achieve higher speeds than those of stan-dard motors has been shown in [57], [58], [59], and [60]. The usage of VSAs in maximizing energy efficiency has been investigated in [61] and in [62]. Some possible applications of speed optimization using VSAs are throwing objects, kicking a ball as a soccer player, or ham-mering a nail. Work in this direction, although in a different setting than what described below, will be reported in [63].
In this paper we address the problem of maximizing link speed at a given position for both constant and variable stiffness actuators (SEA and VSA, respectively) through the application of optimal control theory.
We first present analytical solutions for three SEA cases, consid-ering as control input reference position, speed, and acceleration, re-spectively. The three cases, considered separately, illustrate diverging motivations for the optimal stiffness choice. However, in a realistic setting where the different aspects are merged, we show that an opti-mal (constant) stiffness exists for any given inertia and motor torque. We then study optimal control of VSA and present analytical re-sults illustrating the optimal synchronisation of the spring equilibrium position and stiffness variations. We show that varying the stiffness during the execution of the hammering task improves the final per-formance substantially (a formula for quantifying the improvement is
provided). To demonstrate the realism of assumptions made in the problem setup, and the practical applicability of the obtained control laws, we finally provide experimental results confirming our theoreti-cal predictions.
1.1.1
Problem Definition
In this paper we investigate different optimal control problems which dynamics is always represented by the simplest model of a compliant actuator link system:
¨
q + ω2(q − θ) = 0 , (1.1)
where ω = pk/m and the other variables are defined in Fig. 1.1.
Given a state-form ˙x = f(x, u), and the initial condition x(0) = 0,
our problems mainly differ in the way system state x(t) ∈ Rn and
system inputs u(t) ∈ U ⊂ Rm are defined. Table 1.1 shows the SEA
problems studied, and table 1.2 describes the VSA problem studied.
m
k
q
θ
(a) (b)
Figure 1.1: (a)Scheme of a compliant actuator, where k denotes the spring stiffness, m the link (hammer) inertia, q the link position and θ the rotor position. (b)Picture of the experimental setup
As our goal is always that of maximizing the link speed at the final instant T , we define the performance index
J = φ(x(T )) = x2(T ) = ˙q(T ) . (1.2)
Without lack of generalization, the final position constraint is defined as
ψ(x(T )) = x1(T ) = q(T ) = 0 . (1.3)
The Hamiltonian function is thus reduced to
H(x(t), λ(t), u(t)) = λT(t)f (x(t), u(t)) , (1.4)
where λ(t) ∈ Rn is the vector of the adjoint variables. From the
optimal control theory [64], the necessary conditions to optimize the performance index are:
˙λT (t) = −∂H(x(t), u(t)) ∂x(t) (1.5) λT(T ) = ∂φ(x(T )) ∂x(T ) + ν ∂ψ(x(T )) ∂x(T ) = [ν, 1, 0, . . . , 0] . (1.6)
Given that we are studying autonomous systems without state path constraints, the Hamiltonian is constant. Moreover, in unspecified
terminal time problems where ∂Tφ(x(T )) = 0 and ∂Tψ(x(T )) = 0, we
have the further necessary condition H(x(t), λ(t), u(t))|t=T = 0, hence
we can conclude that:
H(x(t), λ(t), u(t)) = 0 ∀t ∈ [0, T ] . (1.7)
The control domain U is defined as
U = {u : umin < u < umax} , (1.8)
where uminand umaxare the vectors of achievable minimum and
max-imum input values.
Finally, in order to determine the optimal solutions, the Hamilto-nian is maximized along u according to the Maximum Principle [64].
1.1.2
Optimal Control: SEA
In section 1.1.2.1 we investigate the speed optimization of SEAs, [47], without considering path constraints. These are considered in section 1.1.2.2.
1.1.2.1 Models without state path constraints
We tackle the problem considering the following variables as input controls: equilibrium position (P), speed (S), and acceleration (A). Table 1.1 summarizes the most relevant equations to properly expose the adopted method, as described in section 1.1.1. In the first row we present the state space definition of each case. Instead of (1.8)
we can use a simpler inequality constraint: |u| ≤ umax. The second
and the third rows present the Hamiltonian functions, as defined in (1.4), and the co-state dynamics, as defined in (1.5), respectively. The optimal control laws derived with the Maximum Principle are reported
in the fourth row. The switching functions λn(t) as a function of
ν(T ) can be obtained through the solution of the co-state dynamics. By applying the condition (1.7) in t = 0 and by substituting the state initial conditions we can determine the switching function initial condition:
λn(t)|t=0 = 0 . (1.9)
By solving the co-state final values problem and by exploiting (1.9) we obtain ν(T ) as reported in the fifth row of the table 1.1.
We decided to analyze the problem considering one switching only as done by humans when they use their limbs to dash an object, i.e., a soccer ball or to hammer a nail. E.g., during a hammering task, humans change the arm movement direction only once. We assume
that the control value in the first piece is −umax while the second one
is umax (this assumption is proved in section 1.1.3). The solution of
the system of differential equations ˙x = f(x, u) permits us to derive
final state constraint (1.3) to determine the implicit relationship
x1(t1, t)|t=T = 0 . (1.10)
We can also write a second implicit relationship that is determined in correspondence of a zero crossing of the switching function at switch-ing time t = t1:
λn(t)|t=t1 = 0 . (1.11)
At this moment both (1.10) and (1.11) are functions of the system
parameters and the unknowns are t1 and t2 only. If we consider the
suitable change of coordinates
T = t1+ t2, t1 =
c1
ω, t2 =
c2
ω, (1.12)
then (1.10) together with (1.11) compose a system of two equations
and two unknowns, constants c1 and c2, as reported in sixth row
of table 1.1. Note that these constants do not depend on system parameters and can be determined analytically or numerically (results
are reported in table 1.1). Finally t1 and t2 can be obtained from
(1.12) and the link hit speed can be easily evaluated from the solution of x(t) as reported in table 1.1.
1.1.2.2 Constrained model
From the results obtained we can verify that when only θ is con-strained then final speed is proportional to the parameter ω, when only ˙θ is constrained the final speed does not depend on ω and when only ¨θ is constrained the final speed is inversely proportional to ω. We conclude that a realistic analysis of the influence of ω on the fi-nal achievable speed must consider state path constraints on θ, ˙θ, ¨θ concurrently. Thus, in this section we review problem (A) with the further state constraints, |x3| < θmax and |x4| < ˙θmax, where ˙θmax
and θmax represent the maximum motor speed and position
Position control (P) Speed control (S) Acceleration control (A) State space definition xT= [q ˙q] u = θ ˙x = " x2 ω2(u − x 1) # xT= [q ˙q θ] u = ˙θ ˙x = x2 ω2(x3− x1) u xT=hq ˙q θ ˙θi u = ¨θ ˙x = x2 ω2(x 3− x1) x4 u Hamiltonian H = λ1x2+ λ2ω2(u − x1) H = λ1x2+ λ2ω2(x3− x1) + λ3u H = λ1x2+ λ2ω2(x3− x1) + λ3x4+ λ4u Co-state dynamics ˙λ T=hω2λ 2 −λ1 i ˙λT=hω2λ 2 −λ1 −ω2λ2 i ˙λT=hω2λ 2 −λ1 −ω2λ2 −λ3 i Optimal control law u ∗= u
maxsign (λ2) u∗= umaxsign (λ3) u∗= umaxsign (λ4)
Switching function λ2= cos ((T − t) ω) +ν sin ((T − t) ω) ω ν = −ω cot (T ω) λ3= ν − ν cos ((T − t) ω) +ω sin ((T − t) ω) ν = ω sin (T ω) cos (T ω) − 1 λ4= 1 + T ν − tν − cos ((T − t) ω) −ν sin ((T − t) ω)ω ν =ω (−1 + cos (T ω)) T ω − sin (T ) Switching constants ( 1 − 2 cos (c2) + cos (c2+ c1) = 0 csc (c1+ c2) sin (c1) = 0 c1= π , c2= 2 arctan 1 √ 2 ! ( c2− c1− 2 sin (c2) + sin (c2+ c1) = 0
sin (c2) +−1+cos(csin(c1+c1+c2)2)−
cos(c2) sin(c1+c2) −1+cos(c1+c2) = 0 c1= π , c2= π 2 + c2 1+ 2c1c2− c22− 4 cos (c2) +2 cos (c1+ c2) = 0 c1− (c1+ c2) cos (c2) + c2cos (c1+ c2)
+ sin (c1) + sin (c2) − sin (c1+ c2) = 0
c1≈ 2.11 , c2≈ 5.24π
Link final
speed vmax= 2
√
2umaxω vmax= 4umax vmax= 5.74
umax
ω
Table 1.1: Analytically solved optimal control problems for SEA. work and discuss results on the basis of numerical solutions obtained
using the tool ACADO© [65]. The influence of ω on the final speed
is shown in Fig. 1.2 together with the theoretical maximum speeds w.r.t. the problems (P), (S) and (A). It is observed that when position and speed constraints are not active, then system behaves as model (A) (right region of the plot); when the speed constraint is prevalent, then system behaves as model (S) (middle region of the plot); and, when the position constraint is predominant, then system behaves as model (P) (left region of the plot); We thus understand that, given a
0 2 4 6 8 10 0 5 10 15 20 25 30 omega (rad/s)
link final speed (rad/s)
Case [P]+[S]+[A] Case [P] Case [A] Case [S] 0 0.5 1 1.5 2 −10 −5 0 5 10 15 time (s) 0 1 2 3 4 5 −10 −5 0 5 10 time (s) 0 0.2 0.4 0.6 0.8 1 −15 −10 −5 0 5 10 15 time (s) link position link speed control reference position reference speed (*) (*) legend is referred to the three subplots
Figure 1.2: Vmax and hit time T vs ω for cases (P), (S) and (A). The
continuous line shows the behavior of case (A) with boundaries on θmax and ˙θmax.
link inertia, there exists an optimal value of the spring stiffness that maximizes the final achievable speed.
1.1.3
Optimal Control: VSA
The considerations made so far lead us to think that the exploitation of stiffness by a VSA may permit us to work near the optimal speed region (see Fig. 1.2) even for different link inertia. Indeed, we will see that the possibility of adjusting stiffness during the task gives further advantages w.r.t. the SEA case. Thus, in this section we investigate the VSA problem where system inputs are defined as the spring stiffness k and equilibrium position θ.
State space definition xT = [q ˙q] uT = [θ k] ˙x = u2 x2 m(u1− x1) Hamiltonian H = λ1x2− λ2 u2 m(x1− u1) Co-State dynamics ˙λT = " u2 mλ2 − λ1 # λ(T )T = [0 1] Optimal control law u∗ 1 = ( u1,max if λ2 > 0 −u1,max if λ2 < 0 u∗ 2= ( u2,max if λ2(u1− x1) > 0 u2,min if λ2(u1− x1) < 0
Table 1.2: Analytically solved optimal control problems for VSA. The lack of the last three rows shown in table 1.1 is due to the different way to obtain the solutions.
Table 1.2 summarizes the most relevant equation to properly ex-pose the adopted method to analyze the problem, similarly to section
1.1.2. According to (1.8) uT
min =
h
−u1,max u2,min
i =h−θmax kmin i and uT max = h u1,max u2,max i =hθmax kmax i
, where kmax > kmin > 0
and θmax > 0. In this section we first approach the problem without
considering the final position constraint, then we discuss in remark 5 that by shifting the control constraints, the resulting optimal con-trol laws can also be adopted in order to respect a terminal position constraint.
By evaluating ∂uH = 0 we can conclude that u∗must belong to the
boundaries of its domain, and we can obtain the switching conditions
of u∗, reported in the fourth row of table 1.2, by maximizing the
Hamiltonian. The optimal control is a bang-bang control. In the following we present all considerations that allow us to construct the optimal switching sequence.
Remark 1 Given that u1 and u2 are constant between two switching
instants, t′
and t′′
, the solutions of state and co-state dynamics (first
and third row of table 1.2) for t ∈ [t′
, t′′ ] are: x1(t) x2(t) λ1(t) λ2(t) = u1+ (¯x1− u1) cos (ωu2∆t) + ω −1 u2x¯2sin (ωu2∆t) ¯ x2cos (ωu2∆t) + ωu2(u1− ¯x1) sin (ωu2∆t) ¯ λ1cos (ωu2∆t) + ωu2λ¯2sin (ωu2∆t) ¯ λ2cos (ωu2∆t) − ω −1 u2λ¯1sin (ωu2∆t) (1.13) where ωu2 = p
u2/m is the common frequency of all solutions, ¯x1 =
x1(t′), ¯x2 = x2(t′), ¯λ1 = λ1(t′) and ¯λ2 = λ2(t′) are the initial
con-ditions and ∆t = t − t′
. Moreover all the functions are continuous piecewise.
Symbol S2 denotes a switching where u∗2 goes from kmax to kmin while
S1,2 denotes a switching where u∗2 goes from kmin to kmax and u∗1
Theorem 1 The optimal control is characterized by the following properties:
1. the switching sequence is
{S2; S1,2; S2; . . . ; S2; S1,2} , (1.14)
2. the time between S2 and S1,2 is
tS1,2 =
p
m/kminπ/2 , (1.15)
3. the time between S1,2 and S2, the time of the first period and the
time of the last period are: tS2 =
p
m/kmaxπ/2 , (1.16)
To prove theorem 1 we must present some preliminary results. The proof of the theorem is a direct consequence of the following propo-sitions presented in this section. These propopropo-sitions are also compli-mentary in order to have a complete understanding of the optimal control scheme.
Proposition 1 The optimal control u∗
2 at initial time is kmax.
Proof: From the optimal control law, we have that the value of u∗
2|t=0
depends on sign (λ2(u1− x1)) |t=0and that sign(λ2) = sign(u1). Given
that x1|t=0 = 0, then λ2u1|t=0 ≥ 0. Hence, we have the thesis.
Proposition 2 The first switching is S2 and it occurs at time (1.16)
when the speed is piecewise maximum.
Proof: Equation (1.7) evaluated in t = 0 gives λ2|t=0= 0. Consider
(1.13) with ¯λ2 = λ2|t=0, ¯x2 = 0 and, by proposition 1, u2 = kmax.
Since λ2 and u1 − x1 are continuous for t ∈ [0, t′′1], we can find the
two possible switching times by imposing λ2(t′′1) = 0 and |u1,max−
x1(t′′1)| = 0. The latter condition gives the smallest time,i.e., (1.16).
By substituting |u1,max− x1(t′′1)| = 0 in the state dynamics, reported
Proposition 3 The second switching is S1,2 and the time between the
first switching S2 and the second switching S1,2 is given by (1.15).
§ Proof: By substituting ¯λ2 = λ2|t=0, ¯x2 = 0, u2 = kmax in (1.13)
and by evaluating |x(tS2)| we obtain:
|x1(tS2) − u1,max| = 0 (1.17)
|x2(tS2)| = u1,maxωu2,max. (1.18)
By evaluating (1.7) at t = tS2 it follows that λ1|t=tS2 = 0. Since S2
occurs at tS2, by substituting ¯x1 = u1,max, ¯λ1 = λ1|t=tS2 = 0 , u2 =
kmin in (1.13) we obtain x(t) and λ(t) for t ∈ [tS2, t
′′
2]. Since they
are continuous we determine tS1,2 = t
′′
2 − tu2 as the smallest time
between the switchings that can be found by imposing the conditions |x1− u1||t=tS1,2 = 0 and λ2|t=tS1,2 = 0. The latter condition gives the
smallest time, i.e., (1.15). By evaluating |x(tS1,2)| at the instant of
the second switching it follows that:
|x1− u1||t=tS1,2 = |x1− u1||t=0 p (kmax/kmin) = u1,max p (kmax/kmin) (1.19) x2|t=tS1,2 = 0 (1.20)
Since p(kmax/kmin) > 1, then the sign (u1− x1) does not change.
Consequently, u2 changes to kmax because the sign of λ2 changes (as
can be verified in table 1.2). Hence we obtain the thesis.
Remark 2 The proofs of propositions 2 and 3 are independent of the sign of u1.
Proposition 4 The optimal switchings alternate between S2 and S1,2.
The time between S2 and S1,2is given by (1.15), while the time between
Proof: By imposing (1.7) in t = tS1,2 it follows that λ2|t=tS1,2 = 0.
Given (1.20), u2 = kmaxfor t ≥ t′′2 and by proposition 3, we return
to the starting conditions:
x2|t=tS1,2 = x2|t=0 (1.21)
λ2|t=tS1,2 = λ2|t=0= 0 (1.22)
u2|t>tS1,2 = u2|t>0 = kmax (1.23)
Propositions 2 and 3 still hold, and by using logical induction it can be shown that they can be applied recursively. Hence we obtain the thesis.
Proposition 5 At the final time t = T , the optimal control is h
u∗
1(T ) = θmax u∗2(T ) = kmax
i
Proof: The control u∗
1(T ) = θmax comes from λ2(T ) = 1. We prove
that u2(T ) = kmax by contradiction. Assume that the control u∗ is
optimal and u∗
2(T ) = kmin. We have from theorem 4 that the last
switching is S2 and we consider that it happened at tl = T − ∆t. By
propositions 2 and 4 we have that x2(tl) = ˆx2 is maximum on the last
interval in which u1 is constant. Now, if we choose the alternative
control u#(t) = ( 0 if t < ∆t u∗ (t − ∆t) if t > ∆t,
it would have reached the maximum velocity x2(T, u#) = ˆx2 with
maximum stiffness u2(T )#= kmax. Given that x2(T, u#) > x2(T, u∗),
the impossibility is proved.
Remark 3 The number N must be even. This is verified by
contra-diction: if N is odd, then u2at initial time does not respect proposition
1. We can also observe that if N/2 is odd u1|t=0= −u1,max, otherwise
Remark 4 If follows from theorem 1 that: T = π √ m 2 N √kmax+√kmin 2√kminkmax + √1 kmax ! . (1.24)
Remark 5 If −2u1,max ≤ u ≤ 0, then the presented solution
corre-sponds to that of final constraint x1(T ) = 0. This is easy to verify as
follows. When the maximum speed occurs, ¨q = 0 and, consequently,
x1 = u1 (please check table 1.2), i.e., the entire energy of the system
is kinetic and elastic energy stored in the spring must be zero.
There-fore, at final time, x1(T ) = u1,max. If we shift the control contraints
to −2u1,max ≤ u ≤ 0, the theorem 1 still holds, and the presented
so-lution corresponds to that of final constraint x1(T ) = 0. Moreover, it
should be noticed that, fixed δu = u1,max−u1,min, by shifting the control
bounds as described, it is possible to obtain the highest terminal speed, since at the beginning we can stretch the spring with the maximum al-lowed deformation δ, such to store in the system the maximum initial potential energy.
In order to compare maximum speeds obtained when using a SEA
(vSEA) and when using a VSA (vV SA) after a single switching of the
equilibrium position, we have that the final speed after a single switch-ing case is v = 2u1,max p kmax/m 1 + (kmax/kmin)1/2 . (1.25) and consequently: vV SA vSEA = 1 + (kmax/kmin) 1/2 2 . (1.26)
E.g., by assuming kmin = 0.5kmax it is vV SA/vSEA ≈ 1.2. Note that
the comparison bewteen VSA speed and SEA speed is performed
con-sidering the problem with control boundary shifted to [−2umax, 0] in
in (1.25)). Note also that in section 1.1.2 the problem (P) was dis-cussed without considering this bounds’ shifting in order to maintain an uniform presentation of the three cases studied.
We thus conclude that the exploitation of stiffness during the task, improves the performance of the system. Nevertheless we should no-tice that this model, useful to understand the optimal control strategy,
is not very reliable when kmax becomes too high for the same reasons
reported in section 1.1.2. The fact that we neglect the dynamics of the stiffness is subject to the same considerations. Moreover, we implicitly admit that the actuator’s prime movers and the elastic transmission work within a feasible range, and that the achievable stiffness range does not depend on the link and equilibrium positions. Obtained re-sults confirm that with a VSA we can improve the performance of a SEA.
Theorem 2 The stiffness optimal control is:
u2 =
(
u2,max if ˙q¨q > 0
u2,min if ˙q¨q < 0
(1.27)
Proof: From proposition 3 we have that when a S1,2 switching occurs
it is ˙q = 0. From proposition 2 we have that when a S2 switching
occurs it is ¨q = 0. Hence, when a stiffness switching occurs it is ˙q¨q = 0.
From proposition 1 we have that u2|t=t0 = u2,max. Moreover, given
that at initial state ˙q|t=t0 = 0, ¨q|t=t0 = 0 then ˙q¨q > 0|t=t0+.
Hence, as ˙q¨q is a continuous function, we have the thesis.
The optimal control law 1.27 for an hammering example is de-picted in Fig. 1.3.
Figure 1.3: Stiff Speed-up, soft Slow-down the optimal control strategy for a humouring task.
1.1.4
Experimental Results
1.1.4.1 Experimental Setup
The experimental setup comprises one VSA-Cube, a low cost proto-type of a bidirectional antagonistic VSA developed by Centro
Piag-gio [A4], connected to a Simulink interface through an I2C bus. We
mounted two different weights at the edge of the link, which length is
0.15 m. Control inputs are qS = q1+q2 2 and qD = q1
−q2
2 , where q1,2 are
the two motor angles, qS and qD are the output shaft position and
stiffness preset, respectively. An encoder embedded on the actuator reads the link position. By deriving the properly-filtered link position it is possible to obtain link speed.
1.1.4.2 Experimental results
Three experiments were conducted for each mass: in the first one we implemented the optimal control scheme; in the second one we switch only the equilibrium position while keeping the preset constant at its minimum value; and, in the last one, we keep the preset at its maximum value.
re-2 2.1 2.2 2.3 2.4 2.5 2.6 −50 0 50 2 2.1 2.2 2.3 2.4 2.5 2.6 −500 0 500
link speed (deg/s)
2 2.1 2.2 2.3 2.4 2.5 2.6 −50 0 50 time (s) T (a) 2 2.2 2.4 2.6 2.8 −50 0 50 2 2.2 2.4 2.6 2.8 −500 0 500
link speed (deg/s)
2 2.2 2.4 2.6 2.8 −20 0 20 time (s) T (b) 2 2.1 2.2 2.3 2.4 2.5 2.6 −50 0 50 2 2.1 2.2 2.3 2.4 2.5 2.6 −500 0 500
link speed (deg/s)
2 2.1 2.2 2.3 2.4 2.5 2.6 −50
0 50
time (s)
motor 1 position (deg) motor 2 position (deg) link position (deg) link position [filtered] (deg)
position control qs (deg) stiffness control qd (deg) T (c) 3 3.2 3.4 3.6 3.8 −100 0 100 3 3.2 3.4 3.6 3.8 −500 0 500
link speed (deg/s)
3 3.2 3.4 3.6 3.8 −50 0 50 time (s) T (d) 3 3.2 3.4 3.6 3.8 −50 0 50 3 3.2 3.4 3.6 3.8 −500 0 500
link speed (deg/s)
3 3.2 3.4 3.6 3.8 −50 0 50 time (s) T (e) 3 3.2 3.4 3.6 3.8 −100 0 100 3 3.2 3.4 3.6 3.8 −500 0 500
link speed (deg/s)
3 3.2 3.4 3.6 3.8
−50 0 50
time (s)
motor 1 position (deg) motor 2 position (deg) link position (deg) link position [filtered] (deg)
position control qs (deg) stiffness control qd (deg) T
(f)
Figure 1.4: Summary of the experimental results obtained with the VSA-Cube. The two lines are referred to two different masses
mounted on the link. In first column the stiffness preset qD, defined
in [A4], is fixed to the maximum value, while in the second column it’s fixed to the medium or the minimum values. In the third column
qD is an optimized input. Input’s boundary values are qD,min= 5deg,
qD,max = 50deg, qS,min = −20deg and qS,max = 0deg.
sults. It is observed that the first experiment is, indeed, the one that produces the best results. The experiment conducted with low stiff-ness preset presents better performance than that with high stiffstiff-ness preset. We believe that this result is a consequence of the springs’ non-linearity, i.e., the fact that stiffness does not remain constant during the whole experiment. The experimental values of switching times are larger than the theoretically evaluated ones because of the time delays in real system, the limited speed of the VSA-Cube prime movers, and
case type inertia boundary ˙q(T )
(a) SEA-like m = 0.1kg qD,max ≃ 207deg/s
(b) SEA-like m = 0.1kg qD,min ≃ 242deg/s
(c) VSA m = 0.1kg qS,min,max ≃ 276deg/s
(d) SEA-like m = 1kg qD,max ≃ 341deg/s
(e) SEA-like m = 1kg qD,mid ≃ 374deg/s
(f) VSA m = 1kg qS,min,max ≃ 499deg/s
Table 1.3: First and second columns deal with the six experiments’ cases. The third and the fourth show the fixed parameters. Finally, in the fifth column are shown the measured values of link speed at hit time
1.2
Optimality Principles in Stiffness
Con-trol: The VSA Kick
The text of this section is adapted from:
[A5] Garabini M, Passaglia A, Belo F, Salaris P, Bicchi A. Optimality principles in stiffness control: The VSA kick. In: IEEE International Conference on Robotics and Automation (ICRA), 2012.
During the last years the robotics research community has recog-nized that Soft Actuators (Series Elastic Actuators - SEA [47], and Variable stiffness Actuators - VSA) can overcome performance limits of usual actuators. VSAs have been originally designed to improve performances of robots subject to safety constraints [49]. In recent lit-erature SEAs and VSAs also appear in many examples of performance enhancement respect to conventional actuation.
Many works describing the advantageous capabilities of SEAs ( [57–59]), the usage of VSAs in energy efficiency optimization is pre-sented in [61] and in [62]. However, while these solutions shows the better performance of VSAs with respect to conventional actuators, examples in nature show that the true potential of VSAs is still to be explored.
In [49] authors shown that the Optimal Control Theory is an ef-fective tool to understand how to use the stiffness studying a rest to rest position task under a safety constraint in minimum time.
Many applications of performance optimization can be found on our daily life, such as throwing objects, kicking a ball or hammering a nail. Work in this direction was firstly reported in [63] where au-thors present the problem of maximizing final link speed at a specified terminal time and at an unspecified terminal position for SEAs and VSAs. It is shown that the VSA has the best performance. More-over, [66] shows that performance enhancement can be achieved when varying stiffness during dynamic tasks (e.g., throwing a ball). Demon-strations are carried out using numerical simulations and experiments
with a 2 DOFs robotic arm.
In [A3], we tackled the problem of maximizing link velocity at an unspecified terminal time and given final position when using SEAs and VSAs. Analytical results showed that a SEA performing a ham-mering task can reach a speed up to four times that of the actuator’s prime mover when using one equilibrium position switching. We also showed that when position, speed and acceleration constraints are considered, there exists an optimal spring that maximizes the final speed for a given link inertia. A similar result was presented in [67]. In [A3], we also presented an analytical solution for the VSA case, and we demonstrated that the theoretical SEA speed limit can be significantly overcame by a VSA. Experiments confirmed that when adjusting the stiffness during the task it is possible to obtain better performances than using a SEA, showing a speed increase of up to 30%.
In this paper we generalize the work presented in [A3] imposing a new constraint to the problems investigated. Here, we maximize both SEA and VSA link speed not only at a given position but also at a fixed terminal time. In a soccer analogy, this problem formulates an impact optimization of a first-time kick, when the ball has to be hit at a given position in a given time - to be compared with the free terminal time problem, modeling instead a free kick.
This paper is organized as follows. First, we present analytical so-lutions for three SEA cases, considering reference position, speed, and acceleration as control input. The three cases, considered separately, illustrate diverging motivations for the optimal stiffness choice. How-ever, in a realistic setting where the different aspects are merged, we show that an optimal (constant) stiffness exists for any given inertia, terminal time and motor. The main result is that, given a specified fixed time as a problem constraint, the optimal constant stiffness can be obtained as the solution of the unconstrained problem for which the optimal time is coincident to the specified fixed time. This result is discussed by comparing obtained results with the ones presented in our previous paper [A3]. Theoretical results of the SEA problem
m
k
q
θ
(a)
Figure 1.5: Scheme of a compliant actuator, where k denotes the spring stiffness, m the link (hammer) inertia, q the link position and θ the rotor position.
are validated with experimental tests. Second, we study the optimal control of a VSA and present analytical results that illustrate the op-timal synchronization of stiffness and equilibrium position. The opti-mal control policy changes from the unconstrained to the constrained terminal time case is also discussed.
1.2.1
Problem Definition
All the different optimal control problems investigated have dynamics that can be represented by a simple soft actuator model:
¨
q + ω2(q − θ) = 0 , (1.28)
where ω = pk/m and the other variables are defined in figure 1.5.
Given a state-form ˙x = f(x, u), and the initial condition x(0) = 0, our
problems mainly differ in the way system state x(t) = [x1x2 . . . xn]T ∈
Rn and system input u(t) ∈ U ⊂ Rm are defined.
Tables 1.4 and 1.5 detail the fundamental equations that describe the SEA and VSA problems investigated. Given that our objective
is that of maximizing the link speed at the fixed terminal time T , we define the performance index
J = φ(x(T )) = x2(T ) = ˙q(T ) . (1.29)
Without loss of generality, the final position constraint can be defined as
ψ(x(T )) = x1(T ) = q(T ) = 0 . (1.30)
The Hamiltonian function is thus reduced to
H(x(t), λ(t), u(t)) = λT(t)f (x(t), u(t)) , (1.31)
where λ(t) ∈ Rn is the vector of the adjoint variables. From the
optimal control theory [64], the necessary conditions for optimality are: ˙λT (t) = −∂H(x(t), u(t))∂x(t) (1.32) λT(T ) = ∂φ(x(T )) ∂x(T ) + ν ∂ψ(x(T )) ∂x(T ) = [ν, 1, 0, . . . , 0] , (1.33)
where ν is an unknown constant. The control domain U is defined as
U = {u : umin < u < umax} , (1.34)
where uminand umaxare the vectors of achievable minimum and
max-imum input values.
In order to determine the optimal solutions u∗, the Hamiltonian is
maximized along u∗ according to the Maximum Principle [64]. Given
that we are studying autonomous systems without state path con-straints, the Hamiltonian is constant along an optimal solution.
In this paper, as also done in [A3], we analyze the problem con-sidering one switching only (of the reference (equilibrium) position, velocity or acceleration) as done by humans when they use their limbs to dash an object, i.e. a soccer ball or to hammer a nail. E.g. a soc-cer player that perform a first-time kick change the leg movement direction only once.
1.2.2
Optimal Control: SEA
In section 1.2.2.1 we investigate the speed optimization of three dif-ferent models of SEA, without considering state path constraints. In section 1.2.2.2 we tackle the problem for a more realistic SEA model considering also the state path constraints.
1.2.2.1 Models without state path constraints
We consider the following variables as input controls: reference (equi-librium) position (P), speed (S), and acceleration (A). Table 1.4 sum-marizes the most relevant equations to properly expose the adopted method, as described in section 1.2.1. In the first row we present the state space definition for each case. Instead of (1.34), we use a
simpler inequality control constraint |u| ≤ umax. The second and the
third rows present the Hamiltonian functions (1.31), and the co-state dynamics (1.32), respectively. The optimal control laws derived with the Maximum Principle are reported in the fourth row. The
switch-ing functions λn(t), as a function of the unknown constant ν, can be
obtained through the solution of the co-state dynamics. Proposition 6 The optimal control is:
u∗ = ( −umax if 0 ≤ t < t1 umax if t1 < t ≤ T , (1.35)
where t1 ∈ (0, T ) is the switching time of the reference (equilibrium)
position, speed and acceleration (see table 1.4).
Proof: If we consider the optimal control law reported in the fourth
row of table 1.4 and the fact that λn(t)|t=T− > 0 (it follows from
(1.33) and from the co-state dynamics reported in the third row of
table 1.4) we have that u∗
(T ) = umax. Given that we assume one
position switching only, we have the control structure of (1.35). The
and t ∈ (t1, T ]. Then, we apply the final state constraint (1.30) to
determine the implicit relationship
x1(t1, t)|t=T = 0 , (1.36)
and hence the switching time t1.
For the (P) case we show an explicit analytical solution of t1,
unlike in the (S) and (A) cases, for which we report implicit solutions. Finally, the link hit speed can be evaluated from the solution of x(t)
as reported in table 1.4. By imposing λn(t)|t=t1 = 0 and exploiting
the 1.36 we can determine ν.
Figure 1.6: Maximum speed obtainable with position as input (P). In figures 1.6, 1.7 and 1.8 the effect of the final time and system
frequency on achieved Vmax is reported for the three problems. The
upper bound contour of presented plots correspond to SEA maximum velocities of the unconstrained terminal time problem presented in [A3]. From the three figures one may observe that given a terminal time, there exists a frequency that maximizes the final velocity. This
