Design, prototyping and testing of a safety passive system for a Robot with mobile base type "Segway"

University of Pisa

Department of Information Engineering

Master’s degree in

Robotics and Automation Engineering

Design, prototyping and testing

of a safety passive system for a Robot

with mobile base type “Segway”

Candidate

Herrera Alarc´on Edwin Pa´ul

Supervisor:

Prof. Antonio Bicchi PhD. Manolo Garabini

PhD. Manuel Giuseppe Catalano PhD. Giorgio Grioli

Contents

I

Introduction

3

II

Glossary

5

III

Chapters

7

1 State of the Art 7

2 Dynamic Modelling 11

2.1 EGO . . . 11

2.2 Self-balancing Control considering Base Motor Parameters . . . 16

2.2.1 Motor Dynamics . . . 17 2.2.2 LQR 4 states . . . 19 2.2.3 LQR 6 states . . . 21 2.2.4 Control Comparison . . . 23 2.3 Safety System . . . 24 2.3.1 Equivalent Prototype . . . 29 2.4 Variation Prototype . . . 30 3 Simulations 32 3.1 Stability . . . 32

3.1.1 Variation of leg’s mass . . . 32

3.1.2 Variation of initial leg’s angle . . . 33

3.1.3 Variation of leg’s length . . . 34

3.1.4 Variation of leg’s juncture to the body position . . . 36

3.1.5 Variation of spring’s support points position . . . 37

3.1.6 Safety System Parameters . . . 41

3.2 Collisions . . . 42

3.2.1 Surface Considerations . . . 42

3.2.2 Appearance in Generalized Forces Matrix . . . 46

3.3 Adjusting Leg Related Parameters . . . 46

3.4 Disturbances . . . 52

3.5 Vertical Degree of Freedom Analysis . . . 53

3.6 Recovery Working Position . . . 54

3.6.1 Rising . . . 54

3.6.2 Falling Strategy . . . 55

4 System’s design and realization 56 4.1 CAD . . . 56

5 Experimental Validation 59

5.1 Data Validation . . . 59

5.2 Experiment 1: Performing tasks . . . 60

5.2.1 Self-Balancing . . . 60

5.2.2 Moving Arm . . . 61

5.2.3 Reaction to disturbances . . . 62

5.3 Experiment 2: Falling and Rising . . . 62

5.3.1 Falling . . . 62

5.3.2 Rising . . . 64

IV

Conclusions

65

V

Appendix: CAD Drawings

66

Part I

Introduction

EGO is a soft humanoid robot for physical interaction, which currently is used in tele-operation but aims to work with a major level of autonomy in the future. The robot has a soft articulated structure equipped with 12 variable stiffness actuators and two under actuated soft robotic hands developed at the University of Pisa and IIT (SoftHands). The whole system displacement and self-balancing depends on a two-wheeled mobile base, that gives significant mobility to the robot but allocates the working point at its upward vertical state which is unstable.

Control extents an important area in autonomous tasks, however in realistic applications robots need robust and resilient systems to overcome non expected situations. This thesis designs, tests and analyzes a mechanical passive system for two-wheeled robot humanoids equilibrium after losing their balance. The mechanism looks for minimizing as much as possible the consequences given by a collision to the ground.

The first step was to analyze Ego dynamically and the robustness of its self-balancing control. Afterwards, several simulations were done to notice the influence of all the param-eters related to the passive system into the robot’s stability. Followed to the simulation of its collision functionality in case of collapse. This work concluded with construction of a prototype that validates information obtained along simulations. Results corroborate its minimal influence when self-balancing and task performing whether the controller is a LQR or a Dynamic Whole Body Control (DWBC), which considers the whole system dynamics, while smoothing the fall and its potential damages.

Part II

Glossary

wf In superscript means wheel frame

locf In superscript means local frame

bf In superscript means body frame

lf In superscript means leg frame

mw Mass of wheel in Kg

mb Mass of Segway Base in Kg

mr Mass of rod (rest of the body) in Kg

g Gravity in m_s2

Iw Inertia of wheel in Kg m2 (XYZ in subindex denotes axis)

Ib Inertia of base in Kg m2 (XYZ in subindex denotes axis)

Ir Inertia of rod in Kg m2 (XYZ in subindex denotes axis)

R Radius of the wheels in m

w Distance between wheels in m

CoM Center of Mass

l Distance from Segway CoM to Rod CoM in m

l1 Distance from Segway CoM to juncture with leg in m

l2 Distance from Segway CoM to juncture with linear spring in m

h Length of the leg in m

h1 CoM of the leg in m

h2 Distance from juncture of leg and linear spring in m

ωB Angular velocity of the body

φ Rotation angle of the wheels around Y-axis

θ Pitch angle of EGO around Y-axis

ψ Yaw angle of EGO around Z-axis

α Pitch angle of EGO’s support leg around Y-axis

V Voltage given to a motor (w1 or w2 in subindex denotes which wheel)

IM Inertia tensor of the motor

Rm Resistance of the motor

ke Vemf Constant of the motor

kt Torque Constant of the motor

Part III

Chapters

1

State of the Art

Humanoid robotics is a challenging research field, which has received significant attention during the past years and will continue to play an important role in robotics research and in many applications of the 21st century. Ambitious goals have been set for future humanoid robotics. They are expected to serve as companions and assistants for humans in daily life and as ultimate helpers in man-made and natural disasters. In 2050, a team of humanoid robots soccer players shall win against the winner of most recent World Cup. DARPA announced recently the next Grand Challenge in robotics: building robots which do things like humans in a world made for humans. [1]

Considerable progress has been made in humanoid research resulting in a number of humanoid robots able to move and perform well-designed tasks. Over the past decade in humanoid research, an encouraging spectrum of science and technology has emerged that leads to the development of highly advanced humanoid mechatronic systems endowed with rich and complex sensor capabilities. One of the first objectives to reach in humanoid robotics, before thinking in accomplishing different tasks is to obtain balance in a stand up position. Ideally, this state should be maintained while moving (walking, rolling, etc...) as well as while they intend to perform a specific task.

Self-balancing is a study anything but simple. Even if the system by itself seems to be perfectly stable there could be a significant number of external factors, in the work-ing environment, that may inflict changes who can cause repercussions in the system and consequently on stability.

For this reason several companies and laboratories are working hard on robots that are able to stay upright and that are able to stabilize even in the cases where they are introduced disturbances intended to change this state. In fact, robustness is a very important condition to reach in the study of self-balancing.

Even if a perfectly robust system is obtained, there can always be an external condition or situation who can make the robot fall. In fact, lately research rather than tweak the robot’s ability to stand they have decided to approach the problem with a different perspective. Instead of worrying about falling they decided to design a robot that can fall down and get right back up again.

However, falling is not as simple as it seems. As robots are usually kind of fragile it should be a strategy of how to fall correctly or better saying how to reduce the damage produced by a collision on a robot? One of the most used strategies is the implementation of a kind of armor, which covers the interest impact points of the robot. Then, the armor absorbs the impact and then the robot proceeds with to stand up. This can be seen in RHP2, a robot of Tokyo University which was build to endure collision. In fact his body, figure 2, is completely covered by a cage and his posture while falling tries to bend elbows and knees to damp the impact. [2]

Meanwhile, an example of recovery can be seen in figure 3, where there is shown a reliable standing-up routine for a humanoid robot. This particular image is taken from a study done at Freiburg University where is mentioned the importance to detect falls and to implement appropriate recovery procedures. [3] Different algorithms have been studied and implemented to obtain a recovery phase capable of returning the system into a desired position. Nevertheless, an important factor to be considered is the collision by itself who can damage significantly the system.

Figure 3: Robot’s stages to stand up

Some systems use a mechanism similar to a bicycle’s kickstand to hold a stand-up position when they are not used. In figure 4a, an assisted car produced by Murata Manufacturing has a small stick with a wheel at the end. Figures 4b and 4c, show transportation vehicles that use a small rod to hold their weight.

(a) Keepace Assist Car (b) Handlebar Segway (c) Segway Wheelchair Figure 4: Kickstand for two-wheeled systems

This mechanism works pretty well for systems who have most of their mass close to

the wheels. However, if there is a significant mass above from the base, higher safety

considerations must be done to avoid harming the robot.

Figure 5a and 5b, show two-wheeled based robots with a fixed angle passive mechanical mechanism to prevent collisions in the case of unexpected situations. [4] Meanwhile, figure 5c shows Cardea, a project developed by MIT. This system’s particularity is that has an actuated system to deploy kickstands in case some kind of error is detected. [5]

(a) Inverted Pendulum (b) NASA Robonaut RMP (c) MIT’s Cardea

Figure 5: Robust kickstand for two-wheeled systems

To ensure its safety in case of a fall, two active kickstands mounted underneath the RMP optical plate. These failures are usually due to low system batteries or if the RMP encounters a E-stop, a over-inclination error, or some other detectable system error, the Cardea controller generates a digital trigger to deploy the kickstands. The trigger activates a small DC motor that unlatches the spring loaded legs. The legs quickly deploy and latch at 45 degrees, the ballast weight necessary for the stability translates into large forces onto the kickstands during a fall. Consequently, the kickstands are of heavy stainless steel and aluminum construction. [6]

2

Dynamic Modelling

2.1

EGO

The system is analyzed as a 3D model, the simplified model of EGO can be seen as an inverted pendulum in figure 6.

Figure 6: EGO’s simplified model

Euler-Lagrange equations were developed to obtain the dynamic model of this system. The lagrangian can be defined as the subtraction between the kinetic energy of the system

(Ek) and its potential energy (Ep).

L = Ek− Ep d dt ∂L ∂ ˙q  − ∂L ∂q = τ

In figure 7, are shown several frames of reference used to simplify the analysis of the system. It must be mentioned that all calculus are worked in body frame, so the definitive frame of the system is Body Frame.

Velocities from the center of mass of each body (wheels, base and rod) are needed to obtain the dynamic model of the system. Meanwhile, to change frame of reference, the

transformation {}locf _{= R}

Figure 7: EGO’s frames of reference

can be defined as a pure rotational speed, by considering as null the velocity of the contact point against the ground.

vlocf_w1 =     R ˙φ1 0 0     (1) vlocf_w2 =     R ˙φ2 0 0     (2)

In the case there is a yaw rotation from the body around its z-axis, it can be defined as the variation from the wheels rotating at different directions. Noticing that if rotation is the same and with the same orientation (vector sign to be considered) it is equal to zero.

ψlocf =     0 0 R w(φ1− φ2)     ˙ ψlocf =    0 0   

The whole rotation (angular velocity) of Ego in body frame can be composed as its pitch velocity plus the decomposed components of yaw rotation in body frame. There is not considered any roll rotation because it is assumed that it is not a motion plausible for the robot. ωbf_B =     0 ˙ θ 0     + R−1_y (θ) ˙ψlocf =     − ˙ψsin(θ) ˙ θ ˙ ψcos(θ)     (3) v_bbf = v_w1bf + ωbf_B × lw1 to b = R−1y (θ)v locf w1 + ω bf B ×     0 w 2 0     (4) vbf_b =     R 2cos(θ)(φ1+ φ2) 0 R 2sin(θ)(φ1+ φ2)     vbf_r = v_bbf+ ω_Bbf× lb to r = vbfb + ω bf B ×     0 0 l     (5) v_rbf =     R 2cos(θ)(φ1+ φ2) + l ˙θ lR wsin(θ)(φ1− φ2) R 2sin(θ)(φ1+ φ2)    

The components of Ego are supposed to be perfectly symmetric so inertia matrices are diagonal: I =     Ix 0 0 0 Iy 0 0 0 Iz    

To implement Euler-Lagrange equations, kinetic and potential energies of the all Ego’s bodies are explicitly defined in the following equations.

                       Ek w1 = 1₂mwv2w1+ 12Iwω 2 w1 Ek w2 = 1₂mwv2w2+ 12Iwω 2 w2 Ek b = 1₂mbvb2+ 1 2Ibω 2 B Ek r = 1₂mrv2r+ 12Irω 2 B (6)                        Ep w1= mwgR Ep w2= mwgR Ep b = mbgR Ep r = mrg(R + lcos(θ)) (7) Ek= Ek w1+ Ek w2+ Ek b+ Ek r Ep = Ep w1+ Ep w2+ Ep b+ Ep r

The Euler-Lagrange equations are used to obtain manipulator’s equation (8). Where

B(q) is the inertia matrix, C(q, ˙q) is the Coriolis and Centrifugal element matrix, G is the

matrix of gravitational elements, and τ the generalized joint forces.

B(q)¨q + C(q, ˙q) + G(q) = τ (8)

To ease the manipulation of parameters, each matrix component related to the previously mentioned matrices, defined in (9), (10), (11) and (12) are substituted by alphabet numbers with sub-indexes. The letter is related to the matrix it comes from.

B(q) =     b1 b2 b3 b2 b1 b3 b3 b3 Iby+ mrl2     (9) b1 = Iwy+ (mw+ mb+ mr 4 )R 2 + R 2 w2  2Iwz + (Ibz + Irz)cos2(θ) + l2mrsin2(θ) 

b2 = R2 w2  − 2Iwz − (Ibz + Irz)cos2(θ) − l2mrsin2(θ) + w2( mb + mrcos2(θ) + mrsin2(θ) 4 )  b3 = lmrRcos(θ) 2 C(q, ˙q) =     c1 c2 c3     (10) c1 = R ˙θ(Rcos(θ)((−2Ibz − 2Irz+ 2l2mr) ˙φ1+ (2Ibz + 2Irz− 2l2mr) ˙φ2) − lmrw 2_θ)sin(θ)˙ 2 w2 c2 = R ˙θ(Rcos(θ)((2Ibz+ 2Irz− 2l2mr) ˙φ1+ (−2Ibz − 2Irz + 2l2mr) ˙φ2) − lmrw 2_θ)sin(θ)˙ 2 w2 c3 = (R2cos(θ)((Ibz+ Irz− l2mr) ˙φ1 2 + (−2Ibz − 2Irz+ 2l2mr) ˙φ1φ˙2+ (Ibz+ Irz− l2mr) ˙φ2 2 ))sin(θ) w2 G(q) =     0 0 −glmr     (11) τ =     τwheel1 τwheel2 −(τwheel1+ τwheel2)     (12)

In table 1, they are shown Ego’s parameters that are going to be substituted in all dynamics equations regarding the system along this work.

Variable Value mw 1.6 [kg] R 0.13 [m] Iwy 0.0057 [kgm2] Iwz 0.001 [kgm2] mb ≈ 12 [kg] Iby 1.58 [kgm2] mr ≈ 6 [kg] Ibz+ Irz 0.39 [kgm2] w 0.496 [m] l 0.38 [m] g 9.81 [m_s2]

Table 1: Ego parameters

2.2

Self-balancing Control considering Base Motor Parameters

The general problem of designing an optimal control law involves minimizing a cost function which represents an accumulated cost of the sequence of states and control laws from the current discrete time to the final time. For linear systems, this leads to linear state-feedback control, LQR, designed in the next subsection. For nonlinear systems the optimal control problem generally requires a numerical solution, which can be computationally prohibitive.

       ˙x(t) = Ax(t) + Bu(t) y = Cx(t) (13) J = xTSx + Z tf t0 [xTQx + uTRu]dt

This optimal problem is risolved due to Riccati equations resolution.

       ˙ S + SA + AT_{S − SBR}−1_BT_{S + Q = 0} S(tf) = Sf (14)

In order to find a state feedback in the form (15)

Considering it is a problem at infinite horizon, it would make sense if:        limt→∞x(t) = 0 limt→∞u(t) = 0 2.2.1 Motor Dynamics

DC motors are used as the actuator subsystem of EGO. The subsystem directly provides rotary motion and coupled with the wheels allows the robot to make movement. The input to this subsystem is a voltage source applied to the motors armature V , while the output is

torque τw of the gearbox shaft. [7]

Figure 8: Electric equivalent circuit of the armature and rotor diagram

The torque generated by a DC motor is proportional to the armature current and strength of the magnetic field. Assuming the magnetic field is constant, the motor torque is

propor-tional to the armature current by constant factor (kt).

τ = kti

While the shaft is rotating the back emf voltage, Vemf, is generated and is proportional

to the angular velocity of the shaft by constant factor (ke).

      

V = irotorRm+ L∂irotor_∂t + Vemf

τ = IMΦ¨M + bMΦ˙M + τL

(16)

By neglecting the value of coil’s impedance and replacing current from the first equation into the second a direct relationship between voltage and torque is obtained. Taking the first equation of (16):        irotor = V −ke ˙ ΦM Rm τ = ktirotor = _Rk_mt (V − keΦ˙M)

Knowing there is a gearbox between the rotor’s output and the wheel’s torque a gear ratio n is considered. n = τw τL = ˙ ΦM ˙ φW

By substituting these equations in the second equation of (16) the following equation is obtained: kt Rm (V − keΦ˙M) = IMΦ¨M + bMΦ˙M + τL n kt Rm V = IMn2φ¨W +  bM + ktke Rm  n2φ˙W + τw τw = n kt Rm V − IMn2φ¨W −  bM + ktke Rm  n2φ˙W (17)

The relation between angular coordinates is given by ˙φW = ˙φ − ˙θ. This relation is going

to be applied for each motor, meaning ˙φW = ˙φ1− ˙θ and ˙φW = ˙φ2− ˙θ depending on the case.

Variable Value IM 0.10667e-3 [kgm2] Rm 0.84278 [Ω] ke 0.048365 [V _rads ] kt 0.015056 [Nm_A] bM 0.00059249 [N_radms] n 160

2.2.2 LQR 4 states

By introducing an angular displacement variable, (18), the space state defined in previous section, (8), can be reduced into a 2 variable space state where angular displacement and pitch angle are defined.

φ = φ1+ φ2

2 (18)

By adding the first two rows the following expression is obtained:

M = mb + mr B(q) = 2(b1+ b2) 2b3 2b3 Iby+ mrl2 ! C(q, ˙q) = c1 + c2 c3 ! G = 0 −glmr ! τ = τw1+ τw2 −(τw1+ τw2) ! 2Iwy+ R2(2mw+ M ) lmrRcos(θ) lmrRcos(θ) IBy + mrl2 ! ¨_φ ¨ θ ! + −lmrRsin(θ) ˙θ 2 c3− glmrsin(θ) ! = τw1+ τw2 −τw1− τw2 ! (19) τw1+ τw2 = ktn Rm (V1+ V2) − 2IMn2 w R( ¨φ − ¨θ) − ktken2 Rm w R( ˙φ − ˙θ) (20) By replacing (20) in (19), ¯ B = 2Iwy+ R 2_(2m w + M ) + 2IMn2 lmrRcos(θ) − 2IMn2 lmrRcos(θ) − 2IMn2 IBy+ mrl2+ 2IMn2 ! (21) ¯ C =   −lmrsin(θ) ˙θ2 + 2  ktke R  n2( ˙φ − ˙θ) c3− 2  ktke R  n2_{( ˙φ − ˙θ)}   (22)

¯ G = 0 −glmrsin(θ) ! (23) ¯ H = n kt Rm n kt Rm −n kt Rm −n kt Rm ! (24)

A four state linear quadratic controller can be applied from linearizing these equations

by defining: cos(θ) ≈ 1, sin(θ) ≈ θ and ˙θ2 _{≈ 0.}

¯ Blin ¨_φ ¨ θ ! + ¯Clin ˙_φ ˙ θ ! + ¯Glin φ θ ! = ¯HV (25) ¯ Blin = 2Iwy+ R2(2mw + M ) + 2IMn2 lmrR − 2IMn2 lmrR − 2IMn2 IBy+ mrl2+ 2IMn2 ! (26) c3,lin= R2θ(Ibz+ Irz− l2mr)( ˙φ1− ˙φ2)2 ≈ 0 ¯ Clin =   2ktke Rm  n2 ₋₂ktke Rm  n2 −2ktke Rm  n2 ₂ktke Rm  n2   (27) ¯ Glin= 0 0 0 −glmr ! (28) q = φ θ ! ˙ q ¨ q ! = 02x2 I2x2

− ¯BlinG¯lin − ¯BlinC¯lin

! q ˙ q ! + 02x2 ¯ BlinH¯ ! Vw1 Vw2 ! (29) Variable Value Q diag(1,1000,1,1) R diag(0.0008,0.0008)

2.2.3 LQR 6 states

By subtracting the first two equations of (8), a third state variable rotation ψ (yaw angle) can be added to build a 6 state system.

(b1− b2)( ¨φ1− ¨φ2) + (c1− c2) = τw1− τw2 (b1− b2) w R ¨ ψ + (c1− c2) = τw1− τw2 (b1 − b2) w2 2R2ψ + (c¨ 1− c2) w 2R = (τw1− τw2) w2 2R2 b1− b2 = R2 w2  4Iwz+ (Iwy+ mwR2) w2 R2 + 2(Ibz + Irz)cos 2 (θ) + 2l2mrsin2(θ)  (b1− b2)lin = R2 w2  4Iwz+ (Iwy+ mwR2) w2 R2 + 2(Ibz + Irz)  c1− c2 = R2 w2θsin(θ)cos(θ)˙  (−4Ibz− 4Irz+ 4mrl2) ˙ψ  (c1− c2)lin = R2 w2θθ˙  (−4Irz+ 4M l2) ˙ψ  ≈ 0 (b1− b2)lin w2 2R2ψ =¨ w 2R(τw1− τw2) τw1− τw2 = ktn Rm (V1− V2) − IMn2 w R ¨ ψ − ktken 2 Rm w R ˙ ψ b4ψ + c¨ 4ψ = h˙ 4(Vw1− Vw2) (30)

b4 = 2Iwz + mww2 2 + Ibz+ Irz+ (Iwy+ n 2_I M)  w2 2R2  c4 = ktken2 Rm w2 2R2 h4 = w 2R ktn Rm q =     φ θ ψ     ¯ Blin =     2Iwy+ R2(2mw+ M ) + 2IMn2 lmrR − 2IMn2 0 lmrR − 2IMn2 IBy+ mrl2+ 2IMn2 0 0 0 b4     (31) ¯ Clin=     2ktke Rm  n2 ₋₂ktke Rm  n2 ₀ −2ktke Rm  n2 ₂ktke Rm  n2 ₀ 0 0 ktken2 Rm w2 2R2     (32) ¯ Glin =     0 0 0 0 −glmr 0 0 0 0     (33) ¯ H =     n kt Rm n kt Rm −n kt Rm −n kt Rm n kt Rm w 2R −n kt Rm w 2R     (34) ˙ q ¨ q ! = 03x3 I3x3

− ¯BlinG¯lin − ¯BlinC¯lin

! q ˙ q ! + 03x2 ¯ BlinH¯ ! Vw1 Vw2 ! (35) Variable Value Q diag(1,1000,1,1,1,1) R diag(0.0008,0.0008)

2.2.4 Control Comparison

From figure 9, it can be noticed that a 6 states regulator brings better performance to the stability of Ego than the 4 states controller. However, a factor that must be taken in consideration is its linear displacement on the plain which is non null. This controller is going to be implemented along the thesis for self-balancing and also in the case of re-incorporation.

Figure 9: State evolution with initial pitch angle θ0 = 10o

In figure 10, the controller is tested in different initial conditions obtaining a convergence around the equilibrium point. Its convergence in the surroundings of the equilibrium point are clearly noticed.

2.3

Safety System

The system to be analyzed is inspired on a bike’s leg, which rather than simple it completely fulfills its purpose. Following an analog procedure of dynamic analysis, the velocity of this new component is needed to compute Euler-Lagrange equations. A new frame of reference is implemented for this analysis, which can be seen in figure 11. To report its values to body

frame {}b _{= R}

y(α − θ){}l is going to be applied.

Figure 11: Leg frame in Ego

The velocity of the leg is obtained by first knowing the velocity of the juncture point

between body and leg (l1). This point is where the origin of the new frame is placed, to ease

the calculations of the leg’s center of mass velocity.

v_lbf 1 = v bf b + ω bf B × lb to l1 = v bf b + ω bf B ×     0 0 l1     (36)  0

The velocity expression of the center of mass can be reduced into: vbf_h1 = v_lbf₁ + ω_lbf× ll1 to h1 = v bf l1 + Ry(α − θ)ω lf l × ll1to h1 = v bf l1 + Ry(α − θ)ω lf l ×     h1 0 0     (37)

While the energies are grouped in the following expressions:        Ek l = 1₂mlvh21 + 1 2Ilω 2 l Ep l = mrg(R + lcos(θ) − hsin(α)) (38)

Taking Ego’s dynamic equations as basis to design this supportive system, analysis begin from equations: (8), (9), (10), (11) and (12). And then an ultimate row and column is added to include the state of the pitch angle from the leg. The same identification rules are maintained from the previous analysis, and in order to see which factors are included all new terms are defined by a new variable.At first sight it is possible to see the symmetry in

the equations related to φ1 and φ2.

B(q) =       b1+ bα1 b2+ bα1 b3+ bα2 −bα2 b2+ bα1 b1+ bα1 b3+ bα2 −bα2 b3+ bα2 b3+ bα2 Iby+ mrl 2_{+ m} ll12 bα3 −bα2 −bα2 bα3 mlh 2 1       (39) bα1 = mlR2(−2l21+ w2+ 2l21cos(2θ)) 4w2 bα2 = h1mlRsin(α) 2 bα3 = −h1l1mlsin(α − θ) C(q, ˙q) =       c1 + cα1 c2 + cα2 cα3 c       (40)

cα1 =

mlR(−0.5h1w2cos(α) ˙α2+ l1θsin(θ)(l˙ 1Rcos(θ)(2 ˙φ1 − 2 ˙φ2) − 0.5w2θ))˙

w2

cα2 =

mlR(−0.5h1w2cos(α) ˙α2+ l1θsin(θ)(l˙ 1Rcos(θ)(−2 ˙φ1+ 2 ˙φ2) − 0.5w2θ))˙

w2 cα3 = mlw2(−h1l1cos(α − θ) ˙α2− 0.5l21R2sin(2θ)( ˙φ1− ˙φ2)2) w2 cα4 = h1mll1cos(α − θ) ˙θ 2 G(q) =       0 0 −glmr+ gα3 gα4       (41) gα3 = mlw2(−gh1cos(α + θ) − gl1sin(θ)) w2 gα4 = −h1mlgcos(α + θ)

The system of interest is shown in figure 12, where it can be seen the state variables of the system and their influence to the passive subsystem of interest. As well as the design parameters that can variate, always remembering that Ego is already built so its design modifications are limited.

The initial length of the spring is calculated as a trigonometry problem, so its length its defined by the other parameters. This assumption is taken to make parameters defined themselves.

a = θ + π

Figure 12: Reaction force related to the spring b = π − α − θ − π 2 + α = π 2 − θ c = π 2 + θ lspring = q h2 2+ (l1− l2)2− 2h2(l1− l2)cos(a) (42) sin(a) lspring = sin(δ) h2 sin(δ) = h2sin(a) lspring = h2sin(θ + π 2 − α) q h2 2+ (l1− l2)2− 2h2(l1− l2)cos(θ + π₂ − α) (43)

The force is taken from its local frame to the body frame, where all the dynamics of the system are represented.

F_kbf = Ry(δ)Fk =     cos(δ) 0 sin(δ) 0 1 0 −sin(δ) 0 cos(δ)         0 0 Fk     =     Fksin(δ) 0 Fkcos(δ)     (44)

v_lbf 2 = v bf b + ω bf B × lb to l1 = v bf b + ω bf B ×     0 0 l2     (45) vbf_h 2 = v bf l1 + ω bf l × ll1 to h2 = v bf l1 + Ry(α − θ)ω lf l × ll1to h2 = v bf l1 + Ry(α − θ)ω lf l ×     h2 0 0     (46)

By considering the generalized coordinates of the system, the pitch angles of all the bodies

(φ1, φ2, θ, α), the generalized forces can be calculated. This can be done by considering the

velocity at the point where the force is applied and deriving it by the derivative of the generalized coordinates. τi = n X i fi ∂vi ∂ ˙qi

By considering the spring, two forces are going to influence the system. Fk and −Fk

which are given from (44).

τ =       τwheel1 τwheel2

−(τwheel1+ τwheel2) − Fksin(δ)(l1− l2)

Fkh2cos(α − δ − θ)       (47)

2.3.1 Equivalent Prototype

A useful simplification of the proposed prototype is an equivalent system where instead of using a linear spring a torsional one is used. As the spring mass is not considered in the inertia, Coriolis or gravitational matrices, the only matrix that has to be adjusted is the one of the generalized forces.

Figure 13: Torsional Spring Prototype

A torsional spring maintains the exact same relations of a linear one, the only difference is that it considers angle’s positions instead of linear and its derivatives.

τk = Kspring ∗ ((α0− θ0) − (α − θ)) + Kd∗ (− ˙α) τ =       τwheel1 τwheel2 −(τwheel1− τwheel2) − τk τk       (48)

Multi-2.4

Variation Prototype

Considering that part of the collision is going to be absorbed by the passive system, the idea is to use directly its material elasticity for damping. To analyze this problem the leg was simulated as a spring, varying its stiffness to see how it would hold the collision itself.

Figure 14: Linear Spring instead of rigid Leg

Figure 15, shows that leg’s rigidity has to be quite high in order to avoid the system from collision. But exactly having this kind of elasticity module refutes its main hypothesis that was damping the impact to the ground. In this part is exposed an analysis that is lately going to be explained in detail, however it is mentioned know because no further simulations are carried with this prototype.

Figure 15: Reaction falling with a spring as leg

As it is going to be mentioned later, the model of Azad and Featherstone assumes the deformation of one of the bodies related to a collision. Meanwhile, in this case deformation is taken places in both bodies.

3

Simulations

The procedure followed to design Ego’s safety system takes into account the variation of its parameters and influence to self-balancing as well as avoiding collisions impact retail. This approach was considered because Ego is already built and in operation. So that any design decisions have to be done under real physical limitations.

3.1

Stability

As it could be seen from the performance of the controller without the passive system, self-balancing is not ”perfect”. Meaning that the system converges to the unstable equilibrium of interest, nevertheless it oscillates around it. For this reason the following experiment analysis scope is to determine how much influence does this passive system has towards Ego’s stability.

3.1.1 Variation of leg’s mass

From figure 17 and 18, it can be seen that there is a huge variation between 0.4kg and 0.5kg and their behaviour towards self-balancing angle and linear displacement to keep towards an unstable equilibrium point.

Figure 17: Variation of mass of the leg affecting body pitch angle

make sure all the values under it are safe to work with. It can be seen that a frontal weight changes the unstable equilibrium around where the system balances. Bigger the mass, bigger is the displacement of the equilibrium pitch angle.

Figure 18: Variation of mass of the leg affecting displacement

3.1.2 Variation of initial leg’s angle

In 19 and 20, that the initial pitch angle of the leg does not affect deeply the self-balancing angle and displacement.

Figure 19: Variation of leg’s initial position affecting body’s pitch

Nevertheless, from figure 21, it can be seen that with an appropriate choose of the spring stiffness it is possible to ensure leg’s angle. In the figure it can be seen the oscillations that

Figure 20: Variation of leg’s initial position affecting displacement

see that there is not an exactly steady state but it oscillates around a specific point (zero degrees).

Figure 21: Variation of leg’s initial position affecting its evolution

3.1.3 Variation of leg’s length

From figures 22, 23 and 24 it is seen that the length of the leg doesn’t affect significantly the properties of Ego. This is a really important conclusion because it gives liberty design about this parameter. Which in a further section is going to show up important because it would help touch the ground first or even add a contact wheel.

Figure 22: Variation of leg’s length affecting body’s pitch

Figure 23: Variation of leg’s length affecting displacement

3.1.4 Variation of leg’s juncture to the body position

From figures 25, 26 and 27 there is not significant variation in the parameters there have been analyzed until now. As in the case of leg’s length this fact gives liberty in designing procedure without effecting the system’s performance.

The point chosen is 0.3m in the body (≈ 0.43m above the ground) which is close enough to the center of mass but also has enough space for construction because its position is far enough from base motors and its respective electronics.

Figure 25: Variation of leg’s juncture position affecting body’s pitch

Figure 27: Variation of leg’s juncture position affecting its angle.

3.1.5 Variation of spring’s support points position

From the analysis that has been done until now it is possible to notice two important facts about Ego’s safety system parameters. First, the main factor affecting the system is the mass of the leg. Secondly, the importance of spring support points on the body and in the leg. This is because it keeps the leg in the angle of attack (collision angle) of interest and it will retain the body from impact during a fall. There are going to be three analysis done: a variation of each parameter independently and their simultaneous variation.

In figure 28 and 29, pitch angle and displacement do not suffer any particular variations in this simultaneous change of parameters. From figure 30, it is noticed that further from juncture point it reduces leg’s oscillation.

Figure 29: Variation of spring’s juncture positions simultaneously affecting displacement

Figure 30: Variation of spring’s juncture positions simultaneously affecting leg’s angle

Knowing that the best case scenario is the one when spring junctures are far from each other, following variation of leg’s juncture analyzes two cases: a distance between the juncture points of 0.20m and 0.25m. An analog answer from the simultaneous variation experiment is gotten, not significant variations in pitch angle and displacement but an not inconsiderable oscillation from the leg when spring in leg’s juncture is too close to leg’s juncture to the body.

(a) l1− l2= 20cm (b) l1− l2 = 25cm

Figure 31: Variation of spring’s juncture in leg affecting body’s pitch angle

(a) l1− l2= 20cm (b) l1− l2 = 25cm

Figure 32: Variation of spring’s juncture in leg affecting displacement

(a) l1− l2= 20cm (b) l1− l2 = 25cm

At least while talking about linear spring support position, further is better. So the support point on the leg is going to be 0.20m or 0.25m. Noticing as mentioned before that there are physical restraints as motor dimensions, of course without changing leg juncture point. From figure 34, 35 and 36, it is concluded that there are not severe consequences from positioning the spring slightly above the wheels (≈ 0.13m) in 0.15m.

Figure 34: Variation of spring’s juncture in leg affecting pitch body angle

Figure 36: Variation of spring’s juncture in leg affecting leg’s angle

3.1.6 Safety System Parameters

Table 5, resumes the values chosen for for the safety passive system. It must be remembered that these are the very first values for collision simulation and they may be changed depend-ing on need. After obtaindepend-ing several conclusion about design parameters, it is known that

some changes are admitted. As it can be seen l0 is calculated from cosine geometrical law.

This assumption is done to reduce the quantity of independent parameters and to generate a dependence between them.

Variable Value Conditions

α0 π₆ = 30o − l1 0.3 [m] − l2 0.15 [m] − h 0.3 [m] h < R + l1 h1 0.15 [m] − h2 0.25 [m] − kspring 1000N_m = 10_cmN − l0 q (l1− l2)2+ h22− 2(l1− l2)h2cos(θ + π₂ − α) = 0.22[m] −

3.2

Collisions

After analyzing the performance of the passive subsystem towards self-balancing, now it is going to be tested in relation to its objective. To simulate the reaction force from the ground to the body, the ground is modelled as a spring-damper system.

In general, contact normal force models can be classified into two types: rigid and compliant. Rigid models assume that both contacting surfaces remain fully rigid during contact while compliant models assume that at least one of the bodies deforms locally (at the contact area). Given this assumption, the normal force of a compliant model can be expressed as a function of the local deformation and its rate of change.

3.2.1 Surface Considerations

Hunt and Crossley modelled the ground as a nonlinear spring-damper pair at the contact point and introduced a nonlinear equation for the normal force (between a sphere and the ground) as:

F = Kzn+ λzp˙zq

Where z is the penetration distance of the sphere into the ground, both considering undeformed. They chose the values to get similar results to Hertz’s theory:

n = 3

2, p =

3

2, q = 1

A new model is proposed by Azad and Featherstone by defining the contact normal force as the summation of non linear elastic and damping components. The term related to elastic force as is consistent with Hertz’s theory is known to be correct. However, the calculation of damping force has not any strong background for its calculation.

A simplifying assumption is that one material is much harder than the other, so that substantially all of the compression takes place in the softer body. Thus, it is considered that the harder body is rigid and the softer body contains a uniform distribution of infinitely

By defining the force given by each spring and damper as: fKi = Z δi 0 kdξ fDi = b ˙δi

with δi being the deformation of that respective spring-damper pair.

       FK = P∞ i=0fKi = R A Rδi 0 kdξdA FD = P∞ i=0fDi = R Ab ˙δidA (49)

Figure 37: Contact between a sphere and a plate in local deformation and area of contact Where A is the surface of the contact area, l the radius of the contact area and R the radius of the sphere.

Because of the symmetry of deformations towards the center of the contact area (assum-ing its a sphere collid(assum-ing to a plate), all pairs at a distance r from the center have the same deformation. By defining dA = 2πrdr as the surface of a ring element with inner radius r and outer r + dr centered around the center point of the contact area.

       FK = 2π Rl 0r  Rδ(r) 0 kdξ  dr FD = 2π Rl 0rb ˙δ(r)dr (50)

Deformation δ can be calculated and simplified by binomial approximations assuming that r is very small and negligible with respect to R:

δ(r) = −(R − z) +√R2_{− r}2 _{= z −} r

2

Azad and Featherstone took an approach similar to Hunt and Crossley by relating the

model to Hertz’s theory, choosing k(ξ) = βξ−0.5 and d(ξ) = αξ−0.5. Where ξ is the local

deformation of each individual spring-damper pair. By substituting these on current forces:

       FK = 4πβ Rl 0rδ 0.5_(r)dr FD = 2πα Rl 0δ −0.5_{(r) ˙δ(r)dr} (52)

Finally, by replacing local deformation δ(r) a function with a form F = FK + FD is

obtained.        FK = 8₃πβRz 3 2 = K_nz 3 2 FD = 4παRz 1 2 ˙z (53)

Comparing this model to Hertz’s theory,

Kn= 4 3E ∗√ R −→ β = E ∗ 2π√R

as previously mentioned for the damping component there is no theoretical background for its calculation. So its values are taken empirically depending on the model proposed.

The parameter E∗ is determined by a relation of the elasticity moduli and Poisson’s

ratios of the materials regarding the two contacting surfaces.

1 E∗ = 1 − ν₁2 E1 +1 − ν 2 2 E2 (54)

For simulations the materials chosen are a steel leg with spherical end and a surface made of cork. These values are obtained and analyzed by Azad in [8] and [9] so for a deeper investigation of this study, these reports can be seen.

Variable Value

Kn 8.5e6

Dn 3.1e3

R 0.0125 [m]

A comparison of both contact models was done respecting to the system analyzed. From the following images it can be appreciated the reaction of the contact response along with the position of leg’s end. As well as the variation of the spring along collision until stabilization. Following simulations consider Ego’s passive safety system without damping coefficient in its spring. Meaning that all the damping is done by the ground.

It can be seen a better performance by using Azad’s model, where is evident how it manages to describe the damping effect. At the end, this is the factor of interest because due to Hertz’s theorem the elastic component of the force is the same in both cases.

(a) Azad and Featherstone (b) Hunt and Crossley

Figure 38: Comparison Normal Force and Bounce Position

(a) Azad and Featherstone (b) Hunt and Crossley

3.2.2 Appearance in Generalized Forces Matrix

The appearance of normal reaction force from the ground, at least in the case of the leg, has the difference from the other forces in the fact that it appears only when final position of the leg touches the ground. Or as explained previously, in the modelling of the ground section, when this point trespasses the ground.

By having a world frame coupled to a plane that is the ground, a simple comparison to notice where is the end of the leg can be done. So when it is after a threshold, the force appears in the system.

τ =       τwheel1 τwheel2

−(τwheel1+ τwheel2) − Fksin(δ)(l1− l2) − l1N sin(θ)

Fkh2cos(α − δ − θ) − hN cos(α)       (55)

3.3

Adjusting Leg Related Parameters

The first simulation of collision shows that the passive system manages to hold the body of Ego around an angle which is not π. Meaning that the system did not touch the ground. Figure 40, shows the evolution of θ, α and the angle between the body and leg. To avoid crashing of the system there is an angle threshold for body and leg pitch angles. However, this saturation use is going to be avoided as much as possible.

Figure 41, shows the initial position of the spring (blue) and its evolution along collision impact and its damping. Meanwhile, figure 42, shows the contact point position at the end of Ego’s leg and the normal force of the ground which is remembered to be zero when there is not contact.

Figure 41: Spring length evolution

Figure 42: Ground Reaction Force against Leg’s end position

Even if collision simulation went good, the resting pitch angle of Ego’s body produces a contact between its arms and the ground. For further simulations, the length of the leg is increased in 5cm. Here is remembered the importance of previous experimental analysis, leg’s length can be increased without further concerns due to the fact its repercussion to the body is minimum.

Figure 43: Theta variation to the reaction of falling with a saturation in 50o

Figure 44: Alpha variation to the reaction of falling with a saturation in 50o

Figure 45: Ground Reaction Force against Leg’s end position

It is important to notice that depending on the leg’s angle at the moment of collision, determines if it is going to be able to hold Ego’s body from falling to the ground. The best

Figure 46: Pitch angle evolution while falling

Figure 47: Ground Reaction Force against Leg’s end position

In fact it can be noticed that leg’s angle could achieve an angle greater than 90 degrees, which technically will be able to support even though its weight. However, in the case this position is obtained the possibility of rolling in front is highly plausible and this is a situation to avoid.

In order to make much more real the simulation, a damping coefficient must be taken into account when talking about springs.

Figure 48: Reaction falling with a saturation in 50o _{(Damping Included)}

Figure 49: End position of the leg through time

The importance of the damping coefficient can be appreciated directly, due to the abrupt change of velocity after the collision the force expected is bigger than in the case of just stiffness. The damping component softens significantly the collision and motion of the whole system.

By adding a wheel at the end of the leg, the passive system continuous to improve its landing performance. In the case of figure 50, an extra 2.5 cm are added to the leg to reach the objective of maximizing the angle of α, to have the leg as close as possible from the body.

(a) Body’s pitch angle h = 35cm (b) Leg’s pitch angle h = 35cm

(c) Body’s pitch angle h = 37.5cm (d) Leg’s pitch angle h = 37.5cm Figure 50: Comparison leg length with wheel in case h = 35cm and h = 37.5cm

With a much less quantity of parameters the second prototype which uses a torsional spring is simulated. The parameters of Ego’s body are maintained from the last simulation, the only thing that changes is the τ matrix which exposes other kind of generalized force directly relating a momentum.

3.4

Disturbances

Having obtained a proper functional prototype is important to analyze how does the system works and how does it react to external disturbances. Assuming the disturbance is a force acting on Ego’s body, the forces appear in the generalized force matrix, τ , in the following way: τ =        τwheel1+_2wR  FDxwcos(θ) + (2FDylD + FDzw)sin(θ)  τwheel2+_2wR  FDxwcos(θ) + (−2FDylD + FDzw)sin(θ) 

−(τwheel1+ τwheel2) − Fksin(δ)(l1− l2) + FDxlD

Fkh2cos(α − δ − θ)        (56)

The possible disturbance is chosen as a horizontal force that is exerted directly on Ego’s body center of mass. This point is around 40 cm high from the wheel axis.

τ =       τwheel1+1₂FDxRcos(θ) τwheel2+1₂FDxRcos(θ)

−(τwheel1+ τwheel2) − Fksin(δ)(l1− l2) + FDxlD

Fkh2cos(α − δ − θ)       (57)

In figure 52, there is the answer of the system to disturbance (both continue and in-termittent), as it can be seen the system the controller recuperate its position once the disturbance is gone.

(a) Continuous Disturbance (b) Intermittent Disturbance

3.5

Vertical Degree of Freedom Analysis

When modelling the dynamic system of Ego as well as its safety system, there was not done the consideration of overturning. Explicitly there was not considered a degree of freedom regarding motion around the z-axis. To fulfill this experiment, Ego was simulated using Simscape Multibody Library (previously SimMechanics) with which the system is modelled as bodies, joints, constraints, force elements and sensors.

(a) Simulink Block Diagram (b) Body Simulink Block Diagram

Figure 53: Simscape Multibody Library

Contact forces library was used as well, explicitly the sphere to plain contact block. This block was modified in order to simulated ground non-linear models explained in a previous section.

3.6

Recovery Working Position

After damping a fall, it is important that the system returns by its own to a working position. However, there is a certain threshold after whom the controller and the power of the motors won’t be enough to stand-up the system.

Another possible solution is to use Ego’s arms to help the body’s reincorporation. How-ever, this solution depends on the rigidity of the supportive object and the force that the arms can apply while gradually rising the wheels torque to recover the desired position.

3.6.1 Rising

To simulate a standing up returning position, after a free fall, and once the system has stopped, the controller was manually turned on. In figure 55 the whole trajectory is plotted, showing that the system should be able to rise by itself after damping its collision. However, the rise depends on the steady angle which may increase in time due to components of the passive system losing their mechanical properties.

Figure 55: Ground Reaction Force against Leg’s end position

In the case of example, a severe control response can be noticed due to the high weight that has the body’s pitch angle on the self-balancing control. This reaction could make the system to fall backwards, that could be resolved by analogically implement a mechanical

3.6.2 Falling Strategy

In order to avoid strong reactions of the controller to return into equilibrium position may be useful to turn off the controller while falling.

1 2

3

4

Figure 56: Working State-Flow Collision Proposal

The system works normally at the state number 1, the idea is to switch off the controller after Ego’s body pitch angle surpasses certain threshold in angular position and velocity. After that, the system waits until the fall is almost fully damped by the passive system.

State Description

1 Self-Balancing LQR 6 states

2 Falling (No Control)

3 Steady Rest Position

4 Rising (Recovery Control)

As seen in the rising simulation, a better control strategy could be applied for recovering the stand-up position. Mainly focusing in a less aggressive behaviour to avoid constantly damping motion with the passive system aiming to go back to a vertical position.

4

System’s design and realization

Computer-aided design (CAD) was used to create and modify the design of the components related to Ego’s passive system. Due to long waiting periods related to the arrival of com-ponents, a compromising solution was chosen to test the prototype actually working. A hybrid passive system who merged both studied models is used, meaning it has a torsional and linear springs on it. This adaption managed to use components that were at disposal in

that moment in the laboratory. The components available are: torsional springs of ≈ 3N m_rad

and linear springs of ≈ 3000N_m.

Figure 57: Ego base structure modelled in CAD

4.1

CAD

The sizing of components was done around the spring at disposal and some commercial components fast to obtain. These components are an 8 mm bolt of 150 mm of length whose work is being the axis around where the passive system rotates. Meanwhile, the other

rotation axis which is perpendicular to the wheel.

Besides these two main components, several screws M5 and M6 where used within their corresponding nuts and T-nuts. Finally, to hold the linear spring a 30x20 bracket which normally is used to link to perpendicular profiles was used.

(a) Body’s revolute joint 1/2 (b) Body’s revolute joint 2/2 (c) Leg’s revolute joint (d) Caster Wheel-to-leg adapter

(e) Mechanical limita-tion 1/2 (f) Mechanical limita-tion 2/2 (g) Torsional spring holder 1/2 (h) Torsional spring holder 2/2

(i) Alternative spring holder with limitation

(j) Alternative spring holder posed in the leg

Figure 58a and 58b, are assembled using screws KA 25 with conical head, mechanical limits were implemented to fit their holes correctly.

4.2

Leg’s Construction

In figure 59 and 60, there are two views from the safety passive system previous and actual prototype. A new prototype was done because after several experimental repetitions the springs exceeded their elastic deformation region and lost their initial stiffness properties. Meaning that after some trials the passive system was not able to hold the body, but only to reduce the velocity of the system to have a content impact.

(a) Isometric View (b) Front View

Figure 59: Passive System Assembled

Both solutions succeeded its main objective that is damping the fall while avoiding collision that may damage other components. However, second design has a redundancy of springs to avoid the possibility of deformation.

5

Experimental Validation

Considering that the passive mechanical system does not have a gyroscope, a visual method is implemented to analyze the evolution of the angles along with the experiments. To analyze and compare the simulations, information is taken from Ego’s gyroscope and from Kinovea, which is a media player to study motion from videos.

5.1

Data Validation

For the first experiment, tracking the point of interest (juncture between body and leg) was difficult. Because Ego’s long arms oscillate while falling making Kinovea miss the exact point to track. From figure 61, it can be seen that even if the point gets lost, the general behaviour of the system can be appreciated. Specifically, figure 61b, shows a comparison between the dynamic models. It must be mentioned that this experiment was the last before a fatigue failure in the springs.

(a) Video Analysis in Kinovea (b) Comparison of body’s pitch angle Figure 61: Analyzing the collision with hands

By taking off Ego’s hands a much better tracking of points is obtained. Figure 62b, shows a good estimation between a real time sensor and a visual platform to analyze motion from videos. However it can be seen a wide difference between the dynamic model and empirical data. This is because due to limited components and the possibility of extreme fatigue, an

over-design system was implemented with two supportive linear springs to release stress of the spring of interest. However, these springs have an angle different from the main linear spring meaning the force has several components.

(a) Video Analysis in Kinovea (b) Comparison of body’s pitch angle Figure 62: Analyzing the collision without hands

5.2

Experiment 1: Performing tasks

One of the objectives of the falling mechanism is to not limit Ego’s usual tasks as self-balancing or manipulation. These experiments validate what was previously obtained in simulation where the most influential parameter on Ego’s stability is the mass. Still, its influence is almost unscented because of its closeness to the body.

Besides the system guarantees its rigidity due to the stiffness of the springs that make sure the system is not perturbed by a constant oscillation of the passive system.

5.2.1 Self-Balancing

In figure 63b, can be seen an extract of a balancing experiment. Variations are content, keeping an oscillations that goes under 2 degrees. Spring stiffness fix the kickstand and reduce its oscillations while the robot is balancing. In other words, is like the passive system

(a) Ego self-balancing (b) Comparison of angle evolution Figure 63: Stability around unstable equilibrium point

5.2.2 Moving Arm

A much difficult test is to keep balance while a task is performed with Ego’s upper body. The correct execution of this task depends directly on the balance of the system, otherwise, the base abruptly begins to oscillate trying to stabilize the system.

In fact figure 64b, shows bigger oscillations that in the case of just balancing. Neverthe-less, the movement of the the body is still content and its movement corresponds to the one felt on the leg.

(a) Ego moving left arm (b) Comparison of angle evolution Figure 64: Moving Arm

5.2.3 Reaction to disturbances

Keeping the premise that the overall system behaves in standing-up position as a solid body and that constant oscillations of the passive system influence the stability of Ego an experiment consisted in its reactions toward disturbs.

A better perception of disturbs is obtained by the gyroscope, which shows that the leg is only slightly perturbed due to constant pushing.

(a) Ego self-balancing (b) Comparison of angle evolution Figure 65: Stability when pushed around unstable equilibrium point

5.3

Experiment 2: Falling and Rising

These experiments show the accomplishment of the mechanism active objectives. They are defined as active because they actively work to reach a goal while previous objectives focused on being as passive as possible.

5.3.1 Falling

The main goal of the mechanical passive system is to prevent the body from the consequences given from an imminent impact. Both prototypes managed to hold the system from different design composition.

(a) Fully extension of the leg

(b) Reaction from the first extension

(c) Total damping of the fall

(d) Steady position touching the ground Figure 66: Falling Steps First Prototype

Figure 67: Falling Steps Second Prototype

Probably using other rising strategy or with a different type of base motors, a prototype similar to the first one can be used. But to validate simulations the leg was displaced for the next experiment.

5.3.2 Rising

It is a secondary objective that came up while developing the project. Otherwise from its opposite task, depends of a certain pitch threshold to recuperate position just using the two-wheeled base. With angles up to 15 degrees reaches to come back to its working position.

(a) Fully extension of the leg

(b) Reaction from the first extension

(c) Total damping of the fall

(d) Steady position stand up

Figure 68: Rising Steps

Depending on the recovery strategy it may fall backwards. Currently, another passive system on the back of Ego. However, rising may be smooth depending on the recovery control law. Some experiments were done with DWBC that compensates rising with arm movement, releasing less power from the segway base to the system.

Part IV

Conclusions

Ego’s overall system manages to smoothly damp the fall while correctly perform its tasks independent of the controller that is used while it is functioning. Particularly, the dy-namic whole body control confirms the mechanical system’s minimal influence towards self-balancing and manipulation, because it takes into account upper limbs movement.

Losing mechanical properties along testing changes real-time reaction from the passive system. A periodic control of the spring’s stiffness has to be done, to notice if they remain in their elastic zone and ensure the maximal capabilities given by the system.

The first prototype of the passive system weighted approximately 600 grams, remem-bering mass simulations and its influence to self-balancing stability a substantial reduce of weight can improve system’s performance. While, another improvable point is to include the system into Ego’s cover.

Part V

Appendix: CAD Drawings

45.10° 5.50 19.76 13.99 33 33 8.72 12.50 30 16.06 26.87 R4.50 R4.50 5.50 10 12.24 5.50 39.50 46.68

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References

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