Humanoid Robotics: A Reference pp 675-721 | Cite as

# Differential Kinematics

## Abstract

Kinematics play an essential role in motion analysis, motion generation, and control. This chapter focuses on the differential kinematic relations pertinent to open-loop and closed-loop kinematic chains. It is assumed that readers are familiar with the basic concepts concerning systems of rigid bodies, such as representations of positions and orientations, the kinematics of joints, coordinate frame assignment techniques for serial-link and parallel-link mechanisms, coordinate transformations, and the forward and inverse kinematic problems. First, the role of differential kinematics for instantaneous motion analysis is highlighted. The forward and inverse differential kinematic problems for open-loop systems are then formulated using first-order (velocity level) and second-order (acceleration level) relationships. Further on, instantaneous motion analysis and methods for motion generation at special (singular) configurations are discussed. Generic singular configurations for humanoid robots with and without kinematically redundant limbs are highlighted, and the notion of manipulability ellipsoid is introduced. Special attention is paid to solutions to the inverse instantaneous motion problem for kinematic chains with kinematic redundancy. Two basic cases are discussed for redundancy pertinent to a single limb and the whole-body chain. In the latter case, solution methods for multiple-task constraints are highlighted, covering both fixed and variable-priority tasks within a hierarchical structure. Finally, differential kinematic relations within chains with motion constraints stemming from contacts with the environment are derived.

## Notes

### Acknowledgements

The figures were created with the help of Jie Chen and Keita Sagara.

## References

- 1.L. Sciavicco, B. Siciliano,
*Modelling and Control of Robot Manipulators*(Springer Science & Business Media, London, 2000)Google Scholar - 2.R. Featherstone,
*Rigid Body Dynamics Algorithms*(Springer, Boston, 2008)CrossRefGoogle Scholar - 3.J. Denavit, R. Hartenberg, A kinematic notation for lower-pair mechanisms based on matrices. Trans. ASME. J. Appl. Mech.
**22**, 215–221 (1955)Google Scholar - 4.R.M. Murray, Z. Li, S.S. Sastry,
*A Mathematical Introduction to Robotic Manipulation*(CRC Press, Boca Raton, 1994)Google Scholar - 5.J. Yuan, Closed-loop manipulator control using quaternion feedback. IEEE J. Robot. Autom.
**4**(4), 434–440 (1988)CrossRefGoogle Scholar - 6.H.G. Kwatny, G. Blankenship,
*Nonlinear Control and Analytical Mechanics: A Computational Approach*(Springer, New York/Birkhauser, 2000)CrossRefGoogle Scholar - 7.G. Strang,
*Linear Algebra and Its Applications*, Cengage Learning, 4th edn., 19 July 2005Google Scholar - 8.K. Kreutz-Delgado, M. Long, H. Seraji, Kinematic analysis of 7-DoF manipulators. Int. J. Robot. Res.
**11**(5), 469–481 (1992)CrossRefGoogle Scholar - 9.A. Sekiguchi, Y. Atobe, K. Kameta, D. Nenchev, Y. Tsumaki, On motion generation for humanoid robot by the SC approach, in
*Annual Conference of the Robotics Society of Japan*, vol. 2, 2003, p. 2A27Google Scholar - 10.Y. Ogura, H.-O. Lim, A. Takanishi, Stretch walking pattern generation for a biped humanoid robot, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2003, pp. 352–357Google Scholar - 11.R. Kurazume, S. Tanaka, M. Yamashita, T. Hasegawa, K. Yoneda, Straight legged walking of a biped robot, in
*2005 IEEE/RSJ International Conference on Intelligent Robots and Systems*, vol. 2 (IEEE, 2005), pp. 337–343Google Scholar - 12.M. Morisawa, S. Kajita, K. Kaneko, K. Harada, F. Kanehiro, K. Fujiwara, H. Hirukawa, Pattern generation of biped walking constrained on parametric surface, in
*IEEE International Conference on Robotics and Automation*, 2005, pp. 2405–2410Google Scholar - 13.K. Kameta, A. Sekiguchi, Y. Tsumaki, D. Nenchev, Walking control using the SC approach for humanoid robot, in
*IEEE-RAS International Conference on Humanoid Robots*, 2005, pp. 289–294Google Scholar - 14.K. Takahashi, M. Noda, D. Nenchev, Y. Tsumaki, A. Sekiguchi, Static walk of a humanoid robot based on the singularity-consistent method, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2006, pp. 5484–5489Google Scholar - 15.K. Kameta, A. Sekiguchi, Y. Tsumaki, Y. Kanamiya, Walking control around singularity using a spherical inverted pendulum with an underfloor pivot, in
*IEEE-RAS International Conference on Humanoid Robots*, 2007, pp. 210–215Google Scholar - 16.N. Handharu, J. Yoon, G. Kim, Gait pattern generation with knee stretch motion for biped robot using toe and heel joints, in
*IEEE-RAS International Conference on Humanoid Robots*, Daejeon, Korea, 2008, pp. 265–270Google Scholar - 17.Z. Li, N.G. Tsagarikis, D.G. Caldwell, B. Vanderborght, Trajectory generation of straightened knee walking for humanoid robot iCub, in
*IEEE International Conference on Control, Automation, Robotics and Vision*, 2010, pp. 2355–2360Google Scholar - 18.Y. Harada, J. Takahashi, D. Nenchev, D. Sato, Limit cycle based walk of a powered 7DoF 3D biped with flat feet, in
*2010 IEEE/RSJ International Conference on Intelligent Robots and Systems*, Oct 2010, pp. 3623–3628Google Scholar - 19.S. Kotosaka, H. Ohtaki, Selective utilization of actuator for a humanoid robot by singular configuration. J. Robot. Soc. Jpn.
**25**(8), 115–121 (2007)CrossRefGoogle Scholar - 20.H. Arisumi, S. Miossec, J.-R. Chardonnet, K. Yokoi, Dynamic lifting by whole body motion of humanoid robots, in
*2008 IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2008, pp. 668–675Google Scholar - 21.K. Levenberg, A method for the solution of certain non-linear problems in least squares. Q. J. Appl. Math.
**II**(2), 164–168 (1944)MathSciNetCrossRefGoogle Scholar - 22.Y. Nakamura, H. Hanafusa, Inverse kinematic solutions with singularity robustness for robot manipulator control. J. Dyn. Syst. Meas Control
**108**(3), 163 (1986)CrossRefGoogle Scholar - 23.C.W. Wampler, Manipulator inverse kinematic solutions based on vector formulations and damped least-squares methods. IEEE Trans. Syst. Man Cybern.
**16**(1), 93–101 (1986)Google Scholar - 24.T. Sugihara, Solvability-unconcerned inverse kinematics by the Levenberg-Marquardt method. IEEE Trans. Robot.
**27**(5), 984–991 (2011)CrossRefGoogle Scholar - 25.D. Nenchev, Tracking manipulator trajectories with ordinary singularities: a null space-based approach. Int. J. Robot. Res.
**14**(4), 399–404 (1995)CrossRefGoogle Scholar - 26.D. Nenchev, Y. Tsumaki, M. Uchiyama, Singularity-consistent parameterization of robot motion and control. Int. J. Robot. Res.
**19**(2), 159–182 (2000)CrossRefGoogle Scholar - 27.S. Taki, D. Nenchev, A novel singularity-consistent inverse kinematics decomposition for S-R-S type manipulators, in
*IEEE International Conference on Robotics and Automation*, Hong Kong, China, 2014, pp. 5070–5075Google Scholar - 28.Y. Tsumaki, D. Nenchev, S. Kotera, M. Uchiyama, Teleoperation based on the adjoint Jacobian approach. IEEE Control Syst. Mag.
**17**(1), 53–62 (1997)Google Scholar - 29.G.H. Golub, C.F. Van Loan,
*Matrix computations*(Johns Hopkins University Press, Baltimore, 1996)Google Scholar - 30.A.A. Maciejewski, C.A. Klein, The singular value decomposition: computation and applications to robotics. Int. J. Robot. Res.
**8**(6), 63–79 (1989)CrossRefGoogle Scholar - 31.T. Yoshikawa, Manipulability of robotic mechanisms. Int. J. Robot. Res.
**4**(2), 3–9 (1985)Google Scholar - 32.Y. Ogura, H. Aikawa, K. Shimomura, H. Kondo, A. Morishima, H.-O. Lim, A. Takanishi, Development of a new humanoid robot WABIAN-2, in
*IEEE International Conference on Robotics and Automation*, 2006, pp. 76–81Google Scholar - 33.T. Wimbock, D. Nenchev, A. Albu-Schaffer, G. Hirzinger, Experimental study on dynamic reactionless motions with DLR’s humanoid robot Justin, in
*2009 IEEE/RSJ International Conference on Intelligent Robots and Systems*(IEEE, St. Louis, 2009), pp. 5481–5486Google Scholar - 34.K. Kaneko, F. Kanehiro, M. Morisawa, K. Akachi, G. Miyamori, A. Hayashi, N. Kanehira, Humanoid robot HRP-4 – humanoid robotics platform with lightweight and slim body, in
*IEEE International Conference on Intelligent Robots and Systems*, 2011, pp. 4400–4407Google Scholar - 35.I.-W. Park, J.-Y. Kim, J. Lee, J.-H. Oh, Mechanical design of the humanoid robot platform, HUBO. Adv. Robot.
**21**(11), 1305–1322 (2007)CrossRefGoogle Scholar - 36.M. Zucker, S. Joo, M. Grey, C. Rasmussen, E. Huang, M. Stilman, A. Bobick, A general-purpose system for teleoperation of the DRC-HUBO humanoid robot. J. Field Robot.
**32**(3), 336–351 (2015)CrossRefGoogle Scholar - 37.T. Buschmann, S. Lohmeier, H. Ulbrich, Humanoid robot Lola: design and walking control. J. Physiol. Paris
**103**(3–5), 141–148 (2009)CrossRefGoogle Scholar - 38.A. Ben-Israel, T.N. Greville,
*Generalized Inverses – Theory and Applications*. CMS Books in Mathematics, 2nd edn. (Springer, New York, 2003)Google Scholar - 39.A. Liegeois, Automatic supervisory control of the configuration and behavior of multibody mechanisms. IEEE Trans. Syst. Man Cybern.
**7**(12), 868–871 (1977)Google Scholar - 40.Y. Nakamura,
*Advanced Robotics: Redundancy and Optimization*(Addison-Wesley Publishing Company, Reading, 1991)Google Scholar - 41.T. Asfour, R. Dillmann, Human-like motion of a humanoid robot arm based on a closed-form solution of the inverse kinematics problem, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, Las Vegas (IEEE, 2003), pp. 1407–1412Google Scholar - 42.M. Shimizu, H. Kakuya, W.-K. Yoon, K. Kitagaki, K. Kosuge, Analytical inverse kinematic computation for 7-DoF redundant manipulators with joint limits and its application to redundancy resolution. IEEE Trans. Robot.
**24**(5), 1131–1142 (2008)CrossRefGoogle Scholar - 43.R.C. Luo, T.-W. Lin, Y.-H. Tsai, Analytical inverse kinematic solution for modularized 7-DoF redundant manipulators with offsets at shoulder and wrist, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, Chicago, 2014, pp. 516–521Google Scholar - 44.J. Burdick, On the inverse kinematics of redundant manipulators: characterization of the self-motion manifolds, in
*IEEE International Conference on Robotics and Automation*, Scottsdale, 1989, pp. 264–270Google Scholar - 45.D. Whitney, Resolved motion rate control of manipulators and human prostheses. IEEE Trans. Man Mach. Syst.
**10**(2), 47–53 (1969)CrossRefGoogle Scholar - 46.D. Nenchev, Redundancy resolution through local optimization: a review. J. Robot. Syst.
**6**(6), 769–798 (1989)CrossRefGoogle Scholar - 47.B. Siciliano, Kinematic control of redundant robot manipulators: a tutorial. J. Intell. Robot. Syst.
**3**(3), 201–212 (1990)CrossRefGoogle Scholar - 48.T. Yoshikawa, Analysis and control of robot arms with redundancy, in
*First International Symposium on Robotics Research*, Pittsburg (MIT Press, Cambridge, MA, 1994), pp. 735–747Google Scholar - 49.C.A. Klein, B.E. Blaho, Dexterity measures for the design and control of kinematically redundant manipulators. Int. J. Robot. Res.
**6**(2), 72–83 (1987)CrossRefGoogle Scholar - 50.K.L. Doty, C. Melchiorri, C. Bonivento, A theory of generalized inverses applied to robotics. Int. J. Robot. Res.
**12**(1), 1–19 (1993)CrossRefGoogle Scholar - 51.J. Baillieul, Avoiding obstacles and resolving kinematic redundancy, in
*IEEE International Conference on Robotics and Automation*, 1986, pp. 1698–1704Google Scholar - 52.H. Seraji, Configuration control of redundant manipulators: theory and Implementation. IEEE Trans. Robot. Autom.
**5**(4), 472–490 (1989)CrossRefGoogle Scholar - 53.J. Park, W. Chung, Y. Youm, On dynamical decoupling of kinematically redundant manipulators, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 1999, pp. 1495–1500Google Scholar - 54.O. Kanoun, F. Lamiraux, P.-B. Wieber, Kinematic control of redundant manipulators: generalizing the task-priority framework to inequality task. IEEE Trans. Robot.
**27**(4), 785–792 (2011)CrossRefGoogle Scholar - 55.M. Liu, A. Micaelli, P. Evrard, A. Escande, C. Andriot, Interactive virtual humans: a two-level prioritized control framework with wrench bounds. IEEE Trans. Robot.
**28**(6), 1309–1322 (2012)CrossRefGoogle Scholar - 56.M. Brandao, L. Jamone, P. Kryczka, N. Endo, K. Hashimoto, A. Takanishi, Reaching for the unreachable: integration of locomotion and whole-body movements for extended visually guided reaching, in
*IEEE-RAS International Conference on Humanoid Robots*, 2013Google Scholar - 57.A.E. Albert,
*Regression and the Moore-Penrose Pseudoinverse*, 1st edn. (Academic Press, New York, 1972)Google Scholar - 58.M.S. Konstantinov, M.D. Markov, D. Nenchev, Kinematic control of redundant manipulators, in
*11th International Symposium on Industrial Robots*, Tokyo, 1981, pp. 561–568Google Scholar - 59.H. Hanafusa, T. Yoshikawa, Y. Nakamura, Analysis and control of articulated robot arms with redundancy, in
*Prep. of the IFAC ’81 World Congress*, 1981, pp. 78–83.CrossRefGoogle Scholar - 60.D. Nenchev, Restricted Jacobian matrices of redundant manipulators in constrained motion tasks. Int. J. Robot. Res.
**11**(6), 584–597 (1992)CrossRefGoogle Scholar - 61.D. Nenchev, Recursive local kinematic inversion with dynamic task-priority allocation, in
*IEEE International Conference on Robotics and Automation*, Munich, 1994, pp. 2698–2703Google Scholar - 62.B. Siciliano, J.-J. Slotine, A general framework for managing multiple tasks in highly redundant robotic systems, in
*Fifth International Conference on Advanced Robotics*(IEEE, 1991), pp. 1211–1216Google Scholar - 63.G. Antonelli, Stability analysis for prioritized closed-loop inverse kinematic algorithms for redundant robotic systems. IEEE Trans. Robot.
**25**(5), 985–994 (2009)CrossRefGoogle Scholar - 64.H. Sadeghian, L. Villani, M. Keshmiri, B. Siciliano, Dynamic multi-priority control in redundant robotic systems. Robotica
**31**, 1–13 (2013)CrossRefGoogle Scholar - 65.L. Sentis, O. Khatib, Synthesis of whole-body behaviors through hierarchical control of behavioral primitives. Int. J. Humanoid Robot.
**2**(4), 505–518 (2005)CrossRefGoogle Scholar - 66.A.A. Maciejewski, C.A. Klein, Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments. Int. J. Robot. Res.
**4**(3), 109–117 (1985)CrossRefGoogle Scholar - 67.O. Khatib, Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res.
**5**(1), 90–98 (1986)Google Scholar - 68.K. Glass, R. Colbaugh, D. Lim, H. Seraji, Real-time collision avoidance for redundant manipulators. IEEE Trans. Robot. Autom.
**11**(3), 448–457 (1995)CrossRefGoogle Scholar - 69.D. Nenchev, Z. Sotirov, Dynamic task-priority allocation for kinematically redundant robotic mechanisms, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, Munich, 1994, pp. 518–524Google Scholar - 70.O. Brock, O. Khatib, S. Viji, Task-consistent obstacle avoidance and motion behavior for mobile manipulation, in
*IEEE International Conference on Robotics and Automation*, 2002, pp. 388–393Google Scholar - 71.A. Dietrich, T. Wimbock, A. Albu-Schaffer, G. Hirzinger, Reactive whole-body control: dynamic mobile manipulation using a large number of actuated degrees of freedom. IEEE Robot. Autom. Mag.
**19**(2), 20–33 (2012)CrossRefGoogle Scholar - 72.F. Keith, P.-B. Wieber, N. Mansard, A. Kheddar, Analysis of the discontinuities in prioritized tasks-space control under discreet task scheduling operations, in
*IEEE International Conference on Intelligent Robots and Systems*, 2011, pp. 3887–3892Google Scholar - 73.J. Lee, N. Mansard, J. Park, Intermediate desired value approach for task transition of robots in kinematic control. IEEE Trans. Robot.
**28**(6), 1260–1277 (2012)CrossRefGoogle Scholar - 74.T. Petrič, L. Žlajpah, Smooth continuous transition between tasks on a kinematic control level: Obstacle avoidance as a control problem. Robot. Auton. Syst.
**61**(9), 948–959 (2013)CrossRefGoogle Scholar - 75.D. Nenchev, Y. Tsumaki, M. Uchiyama, Real-time motion control in the neighborhood of singularities: a comparative study between the SC and the DLS methods, in
*IEEE International Conference on Robotics and Automation*, 1999, pp. 506–511Google Scholar - 76.N. Mansard, O. Khatib, A. Kheddar, A unified approach to integrate unilateral constraints in the stack of tasks. IEEE Trans. Robot.
**25**(3), 670–685 (2009)CrossRefGoogle Scholar - 77.T. Petrič, A. Gams, J. Babič, L. Žlajpah, Reflexive stability control framework for humanoid robots. Auton. Robots
**34**(4), 347–361 (2013)CrossRefGoogle Scholar - 78.A. Dietrich, T. Wimbock, A. Albu-Schaffer, G. Hirzinger, Integration of reactive, torque-based self-collision avoidance into a task hierarchy. IEEE Trans. Robot.
**28**(6), 1278–1293 (2012)CrossRefGoogle Scholar - 79.H. Sugiura, M. Gienger, H. Janssen, C. Goerick, Real-time collision avoidance with whole body motion control for humanoid robots, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2007, pp. 2053–2058Google Scholar - 80.O. Stasse, A. Escande, N. Mansard, S. Miossec, P. Evrard, A. Kheddar, Real-time (self)-collision avoidance task on a HRP-2 humanoid robot, in
*IEEE International Conference on Robotics and Automation*(IEEE, 2008), pp. 3200–3205Google Scholar - 81.J. Zhao, N.I. Badler, Real time inverse kinematics with joint limits and spatial constraints. Tech. Rep., University of Pennsylvania, 1989Google Scholar
- 82.M. De Lasa, A. Hertzmann, Prioritized optimization for task-space control, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2009, pp. 5755–5762Google Scholar - 83.H. Isermann, Linear lexicographic optimization. OR Spektrum
**4**(4), 223–228 (1982)CrossRefGoogle Scholar - 84.D.E. Stewart, J.C. Trinkle, An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Int. J. Numer. Methods Eng.
**39**(15), 2673–2691 (1996)MathSciNetCrossRefGoogle Scholar - 85.K. Yamane, Y. Nakamura, A numerically robust LCP solver for simulating articulated rigid bodies in contact, in
*Robotics: Science and Systems IV*, Zurich (MIT Press, 2008), pp. 89–104Google Scholar - 86.N. Mansard, F. Chaumette, Task sequencing for high-level sensor-based control. IEEE Trans. Robot.
**23**(1), 60–72 (2007)CrossRefGoogle Scholar - 87.N. Mansard, O. Khatib, Continuous control law from unilateral constraints, in
*IEEE International Conference on Robotics and Automation*, 2008, pp. 3359–3364Google Scholar - 88.N. Mansard, A. Remazeilles, F. Chaumette, Continuity of varying-feature-set control laws. IEEE Trans. Autom. Control
**54**(11), 2493–2505 (2009)MathSciNetCrossRefGoogle Scholar - 89.O. Kanoun, F. Lamiraux, P.-B. Wieber, F. Kanehiro, E. Yoshida, J.-P. Laumond, Prioritizing linear equality and inequality systems: application to local motion planning for redundant robots, in
*IEEE International Conference on Robotics and Automation*, 2009, pp. 2939–2944Google Scholar - 90.A. Escande, N. Mansard, P.-B. Wieber, Fast resolution of hierarchized inverse kinematics with inequality constraints, in
*IEEE International Conference on Robotics and Automation*, 2010, pp. 3733–3738Google Scholar - 91.O. Kanoun, Real-time prioritized kinematic control under inequality constraints for redundant manipulators, in
*Robotics: Science and Systems VII*, ed. by H. Durrant-Whyte, N. Roy, P. Abbeel (MIT Press, Cambridge, MA, 2012), pp. 145–152Google Scholar - 92.D. Montana, The kinematics of contact and grasp. Int. J. Robot. Res.
**7**(3), 17–32 (1988)CrossRefGoogle Scholar - 93.K.H. Hunt, Structural kinematics of in-parallel-actuated robot-arms. J. Mech. Trans. Autom. Des.
**105**(4), 705 (1983)CrossRefGoogle Scholar - 94.A. Dietrich, C. Ott, A. Albu-Schaffer, Multi-objective compliance control of redundant manipulators: Hierarchy, control, and stability, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*(IEEE, 2013), pp. 3043–3050Google Scholar - 95.L. Sentis, J. Petersen, R. Philippsen, Implementation and stability analysis of prioritized whole-body compliant controllers on a wheeled humanoid robot in uneven terrains. Auton. Robots
**35**(4), 301–319 (2013)CrossRefGoogle Scholar - 96.A. Rennuit, A. Micaelli, X. Merlhiot, C. Andriot, F. Guillaume, N. Chevassus, D. Chablat, P. Chedmail, Passive control architecture for virtual humans, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2005, pp. 1432–1437Google Scholar - 97.C. Ott, A. Dietrich, A. Albu-Schaffer, Prioritized multi-task compliance control of redundant manipulators. Automatica
**53**, 416–423 (2015)MathSciNetCrossRefGoogle Scholar - 98.B. Dariush, G.B. Hammam, D. Orin, Constrained resolved acceleration control for humanoids, in
*IEEE/RSJ International Conference on Intelligent Robots and Systems*, 2010, pp. 710–717Google Scholar - 99.G.B. Hammam, P.M. Wensing, B. Dariush, D.E. Orin, Kinodynamically consistent motion retargeting for humanoids. Int. J. Humanoid Robot.
**12**, 1550017 (2015)Google Scholar