Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 6 pps

INSPIRED JUMPING ROBOTJan Babiˇ” c, Damir Omrˇcen and Jadran Lenarˇciˇc Joˇ zef Stefan” Institute, Department of Automatics, Biocybernetics and Robotics Ljubljana, Slovenia jan.babic@ijs

For comparison, a global optimization of different forces acting on themanipulator without null-space techniques should also be considered,with a weighted extended Jacobian approach In addition, automatictechniques for the location of the VEEs should be of interest as well.Future work will also be devoted to add soft-computing techniques forboth trajectory planning and inverse kinematics, and to consider inte-gration with force control on real mobile robot manipulators.

References

Albu-Schaffer, A., Bicchi, A., Boccadamo, G., Chatila, R., De Luca, A., De Santis, A., Giralt, G., Hirzinger, G., Lippiello, V., Mattone, R., Schiavi, R., Siciliano, B., Tonietti, G., Villani, L., “Physical Human-Robot Interaction in Anthropic Do-

mains: Safety and Dependability”, 4th IARP/IEEE-EURON Workshop on nical Challenges for Dependable Robots in Human Environments, Nagoya, J, July

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De Luca, A., “Feedforward/feedback laws for the control of flexible robots” 2000 IEEE International Conference of Robotics and Automation, San Francisco, CA, USA,

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fast and safe motion control”, 11th International Symposium of Robotics Research,

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Zinn, M., Khatib, O., Roth, B., Salisbury, J.K., “A new actuation approach for

hu-man friendly robot design”, International Symposium on Experimental Robotics,

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MA, 1999.

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control”, Robotica, 8, 231–243, 1990.

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Springer-Verlag, London, UK, 2000.

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highly redundant robotic systems, 5th International Conference on Advanced otics, Pisa, I, June 1991.

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De Santis, A., Siciliano, B., Villani, L., “Fuzzy trajectory planning and redundancy

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De Santis, A., Caggiano, V., Siciliano, B., Villani, L., Boccignone, G., “Anthropic

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Featherstone, R., “Resolving manipulator redundancy by combining task constraints”,

Int Meeting Advances in Robot Kinematics, Ljubljana, Yugoslavia, Sep 1988.

Humanoids and Biomedicine

J Babiˇ c, D Omrˇ cen, J Lenarˇ ciˇ c

Modeling time invariance in human arm motion coordination

M Veber, T Bajd, M Munih

Assessment of finger joint angles and calibration of instrumental glove

R Konietschke, G Hirzinger, Y Yan

All singularities of the 9-DOF DLR medical robot setup for minimallyinvasive applications

G Liu, R.J Milgram, A Dhanik, J.C Latombe

On the inverse kinematics of a fragment of protein backbone

V De Sapio, J Warren, O Khatib

Predicting reaching postures using a kinematically constrained shouldermodel

Balance and control of human inspired jumping robot 147

157167177185

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209

INSPIRED JUMPING ROBOT

Jan Babiˇ” c, Damir Omrˇcen and Jadran Lenarˇciˇc

Joˇ zef Stefan” Institute, Department of Automatics, Biocybernetics and Robotics Ljubljana, Slovenia

jan.babic@ijs.si, damir.omrcen@ijs.si, jadran.lenarcic@ijs.si

Abstract The purpose of this study is to describe the necessary conditions for the

motion controller of a humanoid robot to perform the vertical jump.

We performed vertical jump simulations using three different control algorithms and showed the effects of each algorithm on the vertical jump performance We showed that motion controllers which consider one of two conditions separately are not appropriate to control the vertical jump We demonstrated that the motion controller has to satisfy both conditions simultaneously in order to achieve a desired vertical jump.

Keywords:

The vertical jump is an example of a fast explosive movement thatrequires quick and completely harmonized coordination of all segments ofthe robot, for the push-off, for the flight and, finally, for the landing Themost important part of the vertical jump which influences the efficiencyand therefore the height of the jump is the push-off phase The push-offphase can be defined as a time interval when the feet are touching theground before the flight The primary task of the actuators during thepush-off phase is to keep the robot balanced during the entire jump.The secondary task of the actuators is to accelerate the robot’s center

of mass upwards in the vertical direction to the extended body position

In the past, several research groups developed and studied jumpingrobots but most of these were simple mechanisms not similar to humans.They were controlled by empirically derived control strategies Probablythe best-known hopping robots were designed by Raibert, 1986 and histeam They developed different hopping robots, all with telescopic legsand with a steady-state control algorithm Later, De Man et al., 1996developed a trajectory generation strategy based on the angular mo-mentum theorem which was implemented on a model with articulatedlegs Recently Hyon et al., 2003 developed a one-legged hopping robotwith a structure based on the hind-limb model of a dog They used anempirically derived controller based on the characteristic dynamics

Humanoid robot, Vertical jump, dynamic stability

147

© 2006 Springer Printed in the Netherlands.

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 147–156.

sary conditions that the motion controller of a humanoid robot has toconsider in order to perform the vertical jump.

2.

The model of the jumping robot is planar and is composed of foursegments which represent the foot, shank, thigh and trunk (Fig 1).The segments are connected by frictionless rotational hinges whose axesare perpendicular to the sagittal plane The model consists of two parts,the model of the robot in the air and the model of the robot in contactwith the ground While the tip of the foot is on the ground, the contactbetween the foot tip and the ground is modelled as a rotational hingejoint between the foot tip and the ground at point F Therefore, therobot has six degrees of freedom during flight and four degrees of freedomduring stance (with the assumption that the foot tip of the robot doesnot slip and does not bounce back) The generalized coordinates used

to describe the motion of the robot are coordinates x F and y F of the

foot tip measured in the reference frame and joint angles α, β, γ, δ.

Figure 1. Jumping robot during flight.

3.

To assure the verticality of the jump, the robot’s center of mass(COM) has to move in the upward direction above the support poly-gon during the push-off phase of the jump The second condition, which

148

The purpose of this study is to mathematically formulate the

neces-Dynamical Model of Jumping Robot

Vertical Jump Conditions and Control

Algorithm

J Babiþ, D Omrþen and J Lenarþiþ

refers to the balance of the robot during the push-off phase, is the tion of the zero moment point (ZMP) ZMP is the point on the ground

posi-at which the net moment of the inertial forces and the gravity forces has

no component along the horizontal axes (Vukobratovi´c et al., 2004) Inthe following sections we will analyse how these two conditions influencethe vertical jump First we will design two control algorithms based

on the COM condition and ZMP condition separately and then we willdesign a control algorithm that considers both conditions together.Equations that define the position of COM are

where x com and y com are horizontal and vertical positions of COM of the

i and y i are the coordinates of COM of

the i-th segment, m i

g is the quadratic norm of the gravity vector, I i is the inertial tensor of

the i − th segment around its COM and ω i is the angular velocity of the

i −th segment When the robot is at rest, the position of ZMP coincides

with the horizontal position of COM

For the control purposes we have to find the second derivatives of x com and y com (Eq 1) We get the following equations

¨

x com = k11α + k¨ 12β + k¨ 13¨γ + k14¨δ + d1 (4)and

whole system, respectively x

is the mass of the i-th segment and n is the number

of

However, in many cases we can freely move the coordinate system to incide with the position of the desired ZMP and the balancing condition

co-becomes x zmp = 0 In this case we can express x zmp as

x zmp = 0 = k31α + k¨ 32β + k¨ 33¨γ + k34¨δ + d3. (6)Eqs 4, 5 and 6 can be combined and written in the matrix form

where ¨x com and x zmp are the conditions that relate with the balance

On the other hand, ¨y com is the prescribed vertical acceleration of therobot’s COM during the push-off phase of the jump which enables therobot to jump

3.1 Control of x com

In the first case we analyse the vertical jump when the motion troller keeps the horizontal position of the robot’s COM over the virtualjoint connecting the foot with the ground at point F during the entirepush-off phase of the vertical jump Motion controller does not control

The system of equations is determinate and the joint accelerations can

Similarly as in the previous case we have to find the joint accelerations

If we again use the same constraints (9) we get the following determinatesystem of equations

3.3 Control of x com and x zmp

In the third case we will analyse the vertical jump when the motioncontroller considers both conditions from the precedent two sections Itkeeps the position of ZMP and the horizontal position of the robot’sCOM aligned with the virtual joint at point F

In this case the degree of redundancy is one The following constraintthat abolishes the redundancy of Eq 7 is the relationship of the ankleand knee joint accelerations

¨

where C1 is a constant By substitution of Eq 15 into Eq 7 we get

tions, respectively H, C and g denote the inertia matrix, the vector

of Coriolis and centrifugal forces and the vector of gravity forces,

re-spectively ¨q c is the vector of control accelerations ( ¨q c =



¨

α, ¨ β, ¨ γ, ¨ δ

T)

During the push-off phase of the jump ¨q c is defined by Eqs (11),(14) or(17) During the flight phase, when the robot is in the air, the angular

momentum and the linear momentum are conserved and the ¨q c is set insuch a way that the joint motions stops and the robot is prepared forlanding

We performed vertical jump simulations using three different controlalgorithms described in the previous section First we simulated thevertical jump using the control algorithm based on the COM condition,then we simulated the vertical jump using the control algorithm based

on the ZMP condition and, finally, we simulated the jump where thecontroller considered both conditions together

In this case we controlled ¨y comand ¨x com

by Eq 11 From the requirement that ¨x com has to be above the support

com = 0 and ¨x com

the position of COM during the jump

horizontal position while the dashed line represents the vertical position

of COM Dotted line shows the moment of take-off It is evident that

Figure 2. Position of center of mass

during vertical jump considering only the

COM condition.

0 0.2 0.4 0.6 Ŧ6

Ŧ4 Ŧ2 0

Control of In this case we controlled ¨y com and x zmp, as defined

by Eq 14 To satisfy the balance criteria x zmphas to be over the support

polygon (x zmp = 0) As evident from Fig 5, the horizontal position ofCOM during the push-off phase of the jump is not zero and, therefore,the robot does not perform the vertical jump as it should

On the other hand, the torque in the virtual joint is zero (Fig 6) andthe system is balanced without the torque in the virtual joint betweenthe foot and the ground Therefore, the robot performs a jump, but this

is not a vertical jump, since COM is not above point F at the take-off

Control of com

x zmp

x

polygon (point F) follows that x

COM is above point F.the horizontal position of COM remains zero, i.e

The solid line represents the

as defined

= 0 Figure 2 shows

torque at the virtual joint the robot becomes unbalanced Figure 4 shows

moment Figure 7 shows the configurations of the robot during the jump

0.5 1 1.5

x /m

t = 0.13 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x /m

t = 0.25 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x /m

t = 0.38 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

Figure 5. Position of center of mass

during vertical jump considering only

ZMP condition.

0 0.2 0.4 0.6 Ŧ1

Ŧ0.5 0 0.5 1

10 shows theconfigurations of the robot during the jump when the motion controllerconsiders both necessary conditions

J Babiþ, D Omrþen and J Lenarþiþ

the position of ZMP have to coincide with point F Figure

0.5 1 1.5

x /m

t = 0.13 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x /m

t = 0.25 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x /m

t = 0.38 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

Figure 8. Position of center of mass

during vertical jump considering both

COM and ZMP conditions.

0 0.2 0.4 0.6 Ŧ1

Ŧ0.5 0 0.5 1

in vertical jump simulations We showed that motion controllers that

Configurations of robot during vertical jump considering only ZMP

0.5 1 1.5

x/m

t = 0.13 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x/m

t = 0.25 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

x/m

t = 0.38 s

Ŧ0.2 0 0.2 0

0.5 1 1.5

Intel-Raibert M Legged Robots That Balance MIT Press, 1986.

Vukobratovi´ c M and Borovac B Zero-moment point thirty five years of its life.

International Journal of Humanoid Robotics, 1(1):157–173, 2004.

Hyon S., Emura T., and Mita T Dynamics-based control of a one-legged hopping

robot Journal of Systems and Control Engineering, 217(2):83–98, 2003.

J Babiþ, D Omr þþ þen and þþ

–

FOR STABILIZING WHOLE-BODY

MOTIONS OF HUMANOID ROBOTS

Juyong Park and Frank C Park

School of Mechanical & Aerospace Engineering

Seoul National University

juyong.park@gmail.com, fcp@snu.ac.kr

Abstract This paper presents a convex optimization algorithm for the

stabiliza-tion of whole-body mostabiliza-tions for humanoid robots Given a possibly unstable input reference trajectory in the form of joint and base frame acceleration time profiles, the algorithm determines, at each time step, the optimal acceleration profile subject to stability constraints on the zero-moment point (ZMP), and under the assumption that joint posi- tion and velocity measurements are available We show that the above optimization can be formulated as a second-order cone programming (SOCP) problem, a well-known class of convex optimization problem that admits efficient interior-point algorithms Simulations suggest that efficient whole-body stabilization is possible for typical humanoid struc- tures, even in dynamic environments.

Keywords: Whole-body motion, humanoid robot, motion stabilization, convex

op-timization, second-order cone programming

This paper addresses the problem of refining a reference whole-bodymotion for a humanoid robot such that it is stable, and closely approx-imates the reference motion As a possible application scenario, onecan envision a reference motion obtained from human motion capturedata; directly transferring this data to a humanoid robot can easily re-sult in an unstable motion, causing the robot to lose balance We seek

an online algorithm that optimally tracks the reference motion, in anappropriate least-squares sense, while ensuring stability as prescribed

by the zero-moment point (ZMP) condition

Since the early work of [Vukobratovic and Borovac, 2004] on namic stability and stabilization of legged robots using the zero mo-ment point, many methods have been proposed for generation of stablemotions for humanoid robots based on the ZMP notion One of thefirst optimization-based approaches to whole-body motion stabilization

dy-© 2006 Springer Printed in the Netherlands.

157

is the work of Kagami et al [Kagami et al., 2000], who develop an

least square method while satisfying desired ZMP and center-of-gravity(COG) constraints The main disadvantage with this approach is that

COG is constrained from moving along x and y axes in order to simplify

the problem Sugihara and Nakamura [Sugihara et al., 2002] propose analternative COG optimization-based method for balancing a humanoidwith two different loops; this algorithm assumes a stable reference tra-jectory that is subject to short-term disturbances, whereas our objectivepropose a control algorithm for tracking a ZMP trajectory and the mo-tions of some links which want to be controlled This method is usefulfor real-time control, but the resulting ZMP tracking errors can lead tounstable motions Related work preceding the above is [Nishiwakialgorithms for stable motions

In this paper we present a convex optimization algorithm for the bilization of whole-body motions for humanoid robots Given a (possiblyunstable) input reference trajectory in the form of joint and base frameacceleration time profiles, the algorithm determines, at each time step,the optimal acceleration profile subject to stability constraints on thezero-moment point (ZMP), and under the assumption that state mea-

sta-surements (i.e., the joint position and velocity) are available.

We show that the above optimization can be formulated as a order cone programming (SOCP) problem, which is a well-known class

second-of convex optimization problems that admit efficient interior-point rithms Simulation results suggest that online whole-body stabilization

algo-is possible for typical humanoid structures, even in dynamic ments

We assume an n degree-of-freedom humanoid robot with a tree

topol-ogy structure, and define the optimization vector to be

where ˙V0 ∈ se(3) denotes the generalized acceleration of the root link,

and ¨q ∈ n denotes the joint acceleration vector The ensuing strained optimization problem is formulated as

algorithm to achieve dynamic balance for humanoid robots based on the

is to stably adjust an unstable trajectory Park et al [Park et al., 2005]

et al., 2002], [Morisawa et al., 2005], which investigate pattern generation

the coordinates whose origin is at the desired ZMP A V˙i , b V˙i , M ZM P and

C ZM P are functions of position and velocity From these equations thelinear equality constraint (3) follows:

where A V˙ and b V˙ are made by stacking A Vi˙ and b Vi˙ of some links whose

motions need to be constrained (e.g., foot link), and M ZM P,M xy and

where µ is the friction coefficient about the force in the xy plane, µ n is

the rotational friction coefficient about the z axis, F ZM P,f z is the force

along the z axis, F ZM P,f xy is the force in the xy plane, and F ZM P,M z is

the moment about the z axis of F ZM P This problem can be recast as asecond-order cone programming (SOCP) problem by introducing someadditional variables as follows:

the coordinates whose origin is at the desired...

motions need to be constrained (e.g., foot link), and M ZM P,M xy and

where µ is the friction coefficient about the force in the xy plane, µ n... V˙i , M ZM P and< /p>

C ZM P are functions of position and velocity From these equations thelinear equality constraint (3) follows:

where A