The wheeled three-link snake model: singularities in nonholonomic constraints and stick–slip hybrid dynamics induced by Coulomb friction

Abstract

The wheeled three-link snake model is a well-known example of an underactuated robotic system whose motion can be kinematically controlled by periodic changes of its internal shape, coupled with nonholonomic constraints. A known problem of this model is the existence of kinematic singularities at symmetric configurations where the three constraints become linearly dependent. Another critical assumption of this model is that the constraints of zero lateral slippage always hold, which requires large friction at the ground contact. This assumption breaks down when the inputs’ actuation frequency becomes too large, or when passing through singular configurations where the constraint forces grow unbounded. In this work, we extend the kinematic model by allowing for wheels slippage when the constraint forces reach an upper bound imposed by Coulomb friction. Using numerical simulations, we analyze the system’s hybrid dynamics governed by stick–slip transitions at the three wheels. We study the influence of actuation frequency on evolution of stick–slip periodic solutions which induce reversal in direction of net motion, and also show the existence of optimal frequencies that maximize the net displacement per cycle or mean translational speed. In addition, we show that passing through kinematic singularities is overcome by stick–slip transitions which keep the constraint forces and body velocity at finite bounded values. The analysis proves that in some cases, simple kinematic models of underactuated robotic locomotion should be augmented by the system’s hybrid dynamics which accounts for realistic frictional bounds on contact forces.

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Notes

  1. 1.

    Note that for gaits with nonzero net rotation, the net displacement \(\Delta d\) actually depends on the choice of the initial time (phase) of the gait in (9). Nevertheless, for small rotations this dependence has negligible effect, and thus, it is not considered here for simplicity.

References

  1. 1.

    Bloch, A., Baillieul, J., Crouch, P., Marsden, J.E., Zenkov, D., Krishnaprasad, P.S., Murray, R.M.: Nonholonomic Mechanics and Control, vol. 24. Springer, Berlin (2003)

    Google Scholar 

  2. 2.

    Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems, vol. 33. American Mathematical Society, Providence (2004)

    Google Scholar 

  3. 3.

    Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn. 13(4), 369–376 (2008)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Stanchenko, S.: Non-holonomic Chaplygin systems. J. Appl. Math. Mech. 53(1), 11–17 (1989)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bloch, A.M., Krishnaprasad, P., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136(1), 21–99 (1996)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ostrowski, J., Lewis, A., Murray, R., Burdick, J.: Nonholonomic mechanics and locomotion: the snakeboard example. In: Proceedings of 1994 IEEE International Conference on Robotics and Automation, 1994. IEEE, pp. 2391–2397 (1994)

  7. 7.

    Krishnaprasad, P., Tsakiris, D.P.: Oscillations, se(2)-snakes and motion control: a study of the roller racer. Dyn. Syst.: Int. J. 16(4), 347–397 (2001)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bullo, F., Žefran, M.: On mechanical control systems with nonholonomic constraints and symmetries. Syst. Control Lett. 45(2), 133–143 (2002)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chitta, S., Cheng, P., Frazzoli, E., Kumar, V.: Robotrikke: a novel undulatory locomotion system. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005. ICRA 2005. IEEE, pp. 1597–1602 (2005)

  10. 10.

    Chitta, S., Kumar, V.: Dynamics and generation of gaits for a planar rollerblader. In: Proceedings. 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2003.(IROS 2003), vol. 1. IEEE, pp. 860–865 (2003)

  11. 11.

    Kelly, S.D., Murray, R.M.: Geometric phases and robotic locomotion. J Field Robot. 12(6), 417–431 (1995)

    MATH  Google Scholar 

  12. 12.

    Ostrowski, J., Burdick, J.: The geometric mechanics of undulatory robotic locomotion. Int. J. Robot. Res. 17(7), 683–701 (1998)

    Article  Google Scholar 

  13. 13.

    Chakon, O., Or, Y.: Analysis of underactuated dynamic locomotion systems using perturbation expansion: the twistcar toy example. J. Nonlinear Sci. 27(4), 1215–1234 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Shammas, E.A., Choset, H., Rizzi, A.A.: Geometric motion planning analysis for two classes of underactuated mechanical systems. Int. J. Robot. Res. 26(10), 1043–1073 (2007)

    Article  Google Scholar 

  15. 15.

    Nakamura, Y., Ezaki, H., Tan, Y., Chung, W.: Design of steering mechanism and control of nonholonomic trailer systems. IEEE Trans. Robot. Autom. 17(3), 367–374 (2001)

    Article  Google Scholar 

  16. 16.

    Tilbury, D., Murray, R.M., Sastry, S.S.: Trajectory generation for the N-trailer problem using Goursat normal form. IEEE Trans. Autom. Control 40(5), 802–819 (1995)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hatton, R.L., Choset, H.: Geometric swimming at low and high Reynolds numbers. IEEE Trans. Robot. 29(3), 615–624 (2013)

    Article  Google Scholar 

  18. 18.

    Alouges, F., DeSimone, A., Giraldi, L., Zoppello, M.: Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers. Int. J. Non-Linear Mech. 56, 132–141 (2013)

    Article  Google Scholar 

  19. 19.

    Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15(4), 255–289 (2005)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Miloh, T., Galper, A.: Self-propulsion of general deformable shapes in a perfect fluid. Proc. R. Soc. Lond. A 442(1915), 273–299 (1993)

    Article  Google Scholar 

  21. 21.

    Gutman, E., Or, Y.: Symmetries and gaits for Purcell’s three-link microswimmer model. IEEE Trans. Robot. 32(1), 53–69 (2016)

    Article  Google Scholar 

  22. 22.

    Melli, J.B., Rowley, C.W., Rufat, D.S.: Motion planning for an articulated body in a perfect planar fluid. SIAM J. Appl. Dyn. Syst. 5(4), 650–669 (2006)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kanso, E., Marsden, J. E.: Optimal motion of an articulated body in a perfect fluid. In: 44th IEEE Conference on Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC’05. IEEE, pp. 2511–2516 (2005)

  24. 24.

    Tam, D., Hosoi, A.E.: Optimal stroke patterns for Purcell’s three-link swimmer. Phys. Rev. Lett. 98(6), 068105 (2007)

    Article  Google Scholar 

  25. 25.

    Wiezel, O., Or, Y.: Using optimal control to obtain maximum displacement gait for Purcell’s three-link swimmer. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp. 4463–4468 (2016)

  26. 26.

    Murray, R.M., Li, Z., Sastry, S.S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1994)

    Google Scholar 

  27. 27.

    Sidek, N., Sarkar, N.: Dynamic modeling and control of nonholonomic mobile robot with lateral slip. In: Third International Conference on Systems: ICONS 08. IEEE. pp. 35–40 (2008)

  28. 28.

    Bazzi, S., Shammas, E., Asmar, D., Mason, M.T.: Motion analysis of two-link nonholonomic swimmers. Nonlinear Dyn. 89(4), 2739–2751 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Salman, H., Dear, T., Babikian, S., Shammas, E., Choset, H.: A physical parameter-based skidding model for the snakeboard. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp. 7555–7560 (2016)

  30. 30.

    Tian, Y., Sidek, N., Sarkar, N.: Modeling and control of a nonholonomic wheeled mobile robot with wheel slip dynamics. In: IEEE Symposium on Computational Intelligence in Control and Automation: CICA 2009. IEEE, pp. 7–14 (2009)

  31. 31.

    Dear, T., Kelly, S. D., Travers, M., Choset, H.: Snakeboard motion planning with viscous friction and skidding. In: 2015 IEEE International Conference on Robotics and Automation (ICRA). IEEE, pp. 670–675 (2015)

  32. 32.

    Fedonyuk, V., Tallapragada, P.: Stick–slip motion of the Chaplygin sleigh with a piecewise-smooth nonholonomic constraint. J. Comput. Nonlinear Dyn. 12(3), 031021 (2017)

    Article  Google Scholar 

  33. 33.

    Cheng, P., Frazzoli, E., Kumar, V.: Motion planning for the roller racer with a sticking/slipping switching model. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, 2006. ICRA 2006. IEEE, pp. 1637–1642 (2006)

  34. 34.

    Zadarnowska, K., Oleksy, A.: Motion planning of wheeled mobile robots subject to slipping. J. Autom. Mob. Robot. Intell. Syst. 5, 49–58 (2011)

    Google Scholar 

  35. 35.

    Tarakameh, A., Shojaei, K., Shahri, A. M.: Adaptive control of nonholonomic wheeled mobile robot in presence of lateral slip and dynamic uncertainties. In: 2010 18th Iranian Conference on Electrical Engineering (ICEE). IEEE, pp. 592–598 (2010)

  36. 36.

    Tian, Y., Sidek, S.N., Sarkar, N.: Tracking control for nonholonomic wheeled mobile robot with wheel slip dynamics. In: ASME: Dynamic Systems and Control Conference. American Society of Mechanical Engineers 2009, pp. 739–746 (2009)

  37. 37.

    Nandy, S., Shome, S., Somani, R., Tanmay, T., Chakraborty, G., Kumar, C.: Detailed slip dynamics for nonholonomic mobile robotic system. In: 2011 International Conference on Mechatronics and Automation (ICMA). IEEE, pp. 519–524 (2011)

  38. 38.

    Transeth, A.A., Pettersen, K.Y., Liljebäck, P.: A survey on snake robot modeling and locomotion. Robotica 27(7), 999–1015 (2009)

    Article  Google Scholar 

  39. 39.

    Krishnaprasad, P. S., Tsakiris, D. P.: G-snakes: nonholonomic kinematic chains on lie groups. In: Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, vol. 3. IEEE, pp. 2955–2960 (1994)

  40. 40.

    Matsuno, F., Suenaga, K.: Control of redundant 3D snake robot based on kinematic model. In: Proceedings of IEEE International Conference on Robotics and Automation, 2003. ICRA’03, vol. 2. IEEE, pp. 2061–2066 (2003)

  41. 41.

    Matsuno, F., Sato, H.: Trajectory tracking control of snake robots based on dynamic model. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation, 2005. ICRA 2005. IEEE, pp. 3029–3034 (2005)

  42. 42.

    Tanaka, M., Tanaka, K.: Singularity analysis of a snake robot and an articulated mobile robot with unconstrained links. IEEE Trans. Control Syst. Technol. 24(6), 2070–2081 (2016)

    Google Scholar 

  43. 43.

    Dear, T., Kelly, S.D., Travers, M., Choset, H.: The three-link nonholonomic snake as a hybrid kinodynamic system. In: American Control Conference (ACC). IEEE 2016, pp. 7269–7274 (2016)

  44. 44.

    Dear, T., Kelly, S.D., Travers, M., Choset, H.: Locomotive analysis of a single-input three-link snake robot. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp. 7542–7547 (2016)

  45. 45.

    Wiezel, O., Or, Y.: Optimization and small-amplitude analysis of Purcell’s three-link microswimmer model. Proc. R. Soc. A 472(2192), 20160425 (2016)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Wendel, E. D., Ames, A. D.: Rank properties of Poincaré maps for hybrid systems with applications to bipedal walking. In: Proceedings of the 13th ACM international conference on Hybrid systems: computation and control. ACM, pp. 151–160 (2010)

  47. 47.

    Gamus, B., Or, Y.: Dynamic bipedal walking under stick–slip transitions. SIAM J. Appl. Dyn. Syst. 14(2), 609–642 (2015)

    MathSciNet  Article  Google Scholar 

  48. 48.

    Gutman, E., Or, Y.: Optimizing an undulating magnetic microswimmer for cargo towing. Phys. Rev. E 93(6), 063105 (2016)

    Article  Google Scholar 

  49. 49.

    Shampine, L.F., Gear, C.W.: A user’s view of solving stiff ordinary differential equations. SIAM Rev. 21(1), 1–17 (1979)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Springer, Berlin (2008)

    Google Scholar 

  51. 51.

    Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications, vol. 163. Springer, Berlin (2008)

    Google Scholar 

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Appendices

Appendix A: Dimensional analysis of the dynamics and scaling with actuation frequency

Consider the constrained dynamical system in (14), combined with the time derivative of the constraints in (15). The body acceleration can be obtained from (14) as

$$\begin{aligned} \ddot{{\mathbf q}}_b={\mathbf {M}}_{bb}^{-1}\left( {\mathbf {W}}_b^T {{\varvec{\Lambda }}}- {\mathbf {M}}_{bs} \ddot{{\mathbf q}}_s - {\mathbf {B}}_b \right) . \end{aligned}$$
(22)

Substituting (22) into (15), an explicit expression for the vector \({{\varvec{\Lambda }}}\) of constraint forces is obtained as:

$$\begin{aligned} \begin{array}{l} {{\varvec{\Lambda }}}= \left( {\mathbf {W}}_b {\mathbf {M}}_{bb}^{-1} {\mathbf {W}}_b^T \right) ^{-1} \Big ({\mathbf {W}}_b {\mathbf {M}}_{bb}^{-1} \big ({\mathbf {M}}_{bs} \ddot{{\mathbf q}}_s + {\mathbf {B}}_b \big )\\ \qquad \;\; - {\mathbf {W}}_s \ddot{{\mathbf q}}_s - \dot{{\mathbf W}}_b \dot{{\mathbf q}}_b \Big ) \end{array} \end{aligned}$$
(23)

Next, consider a periodic gait input \({\mathbf {q}}_b(t)\) of form (9), with a variable actuation frequency \(\omega \). The shape velocity and acceleration vectors thus scale as \(\dot{{\mathbf q}}_s \sim \omega \) and \(\ddot{{\mathbf q}}_s \sim \omega ^2\). Assuming that the gait does not cross any singularity \(\det ({\mathbf {W}}_b) \ne 0\), the kinematic relation (6) implies that the body velocity scales as \(\dot{{\mathbf q}}_b \sim \omega \), and (15) implies that \(\ddot{{\mathbf q}}_b \sim \omega ^2\). Finally, the velocity-dependent vector \({\mathbf {B}}_b({\mathbf {q}},\dot{{\mathbf q}})\) and matrices \(\dot{{\mathbf W}}_b,\dot{{\mathbf W}}_s\) are quadratic in the velocities \(\dot{{\mathbf q}}\) and thus scale as \(\omega ^2\). Therefore, (23) implies that the vector \({{\varvec{\Lambda }}}\) of constraint forces scales as \(\omega ^2\).

Appendix B: Singularity analysis of the hybrid dynamics

We now analyze the conditions for singularity in the dynamics of all possible combinations of stick–slip states. The dynamic equations of motion are given in (14), which gives five scalar equations. These equations are augmented by (20), giving rise to a \(8 \times 8\) linear system of the form \({{\mathbf {A}}}{{\mathbf {z}}}= {\mathbf {b}}\) where the vector of unknowns is \({{\mathbf {z}}}=({\ddot{{\mathbf q}}_b,{{\varvec{\tau }}},{{\varvec{\Lambda }}}})\). This can be further simplified by direct substitution of the constraint force \({\lambda }_i\) for each slipping wheel from (20) and eliminating \({\lambda }_i\) from the vector \({{\mathbf {z}}}\). This reduces the dimension of the linear system to \(8-n_s\) where \(n_s\) is the number of slipping wheels. We then calculate the determinant D of the linear system’s \((8-n_s)\times (8-n_s)\) matrix and analyze the conditions for \(D=0\), which is a singularity of the dynamics. In the following, we repeat this process for all possible combinations of stick–slip states, while disregarding symmetries between the two side links \(i=1,2\). We use the abbreviations \(s_i=\sin \phi _i,\; c_i=\cos \phi _i\) for \(i=1,2\). First, consider the case where only wheel 1 is slipping. The determinant D is obtained as:

$$\begin{aligned} \begin{array}{l} D=\frac{m_1 l^2}{3} \left( 3\eta -6s_1 s_2 -2 c_2^2 +2\eta ^3 c_2^2 +6\eta c_1 +12\eta c_2 \right. \\ \left. \; \qquad +6\eta ^2+\eta ^3 +6\eta ^2 c_2 -6\eta c_1 c_2^2 -6\eta c_2 s_1 s_2+8\right) . \end{array} \end{aligned}$$

It can be shown that the minimal value of D is zero, which is attained only for \(\eta = 1\) and \(\phi _2 = \pm \, \pi \). These values of joint angle are nonphysical, due to collision between links and coincidence of wheels 0 and 2.

Second, consider the case where only wheel 2 is slipping. The determinant D is obtained as:

$$\begin{aligned} \begin{array}{l} D=\frac{m_1 l^2}{3}(6\eta +2 \cos (2\phi _1+2\phi _2 )+12 \cos (\phi _1+\phi _2 )\\ \qquad +\,3\eta ^2 \cos (\phi _1+2\phi _2 )+3\eta ^2 \cos (2\phi _1+\phi _2 )+6\eta ^2 \cos (2\phi _1 ) \\ \qquad +\,6\eta ^2 \cos (2\phi _2 )+3\eta ^3 \cos (2\phi _1)+3\eta ^3 \cos (2\phi _2 ) \\ \qquad +\,6\eta \cos (\phi _1+\phi _2 ) +\eta ^3 \cos (2\phi _1+2\phi _2 )+18\eta \cos (\phi _1 ) \\ \qquad +\,18\eta \cos (\phi _2 )+12\eta ^2+5\eta ^3+6\eta \cos (\phi _1+2\phi _2 ) \\ \qquad +\,6\eta \cos (2\phi _1+\phi _2 )+9\eta ^2 \cos (\phi _1 )+9\eta ^2 \cos (\phi _2 )+10) \end{array} \end{aligned}$$

Setting \(D=0\), it can be shown that singularity occurs when \(\phi _1+\phi _2 = \pm \,\pi \) or when \(\phi _1,\phi _2 = \pm \,\pi \).

Next, consider the case where wheels 1 and 2 are both slipping. The determinant D is obtained as:

$$\begin{aligned} \begin{array}{l} D=\frac{m_1^2 l^2}{3} \left( 8\eta -6s_1s_2+3c_1^2+3c_2^2+12\eta c_1+12 \eta c_2 \right. \\ \left. \qquad +12\eta ^2+8\eta ^3+\eta ^4+6\eta ^2 c_1+6\eta ^2 c_2+10 \right) . \end{array} \end{aligned}$$

The minimal value of \(D_\mathrm{{min}}=\frac{4 m_1^2 l^2}{3}>0\) is obtained in the nonphysical case of \(\eta =0\), for \(\phi _1 = \phi _2 = \pi /2\).

Fig. 13
figure13

Time plots of simulation results under torque input: a joint angles \(\phi _i(t)\), b joint velocities \(\pm \, \dot{\phi }_i(t)\), c norm of velocity \(||\dot{{\mathbf q}}(t)||\)

Now consider the case where wheels 0 and 1 are both slipping. The determinant D is obtained as:

$$\begin{aligned} \begin{array}{l} D=\frac{m_1^2 l^2}{6} \left( 28\eta +3\cos (2\phi _1+2\phi _2)+24\cos (\phi _1+\phi _2) \right. \\ \qquad +\,6\eta ^2 \cos (\phi _1+2\phi _2) +12\eta ^2 \cos (2\phi _2)+12\eta ^3 \cos (2\phi _2) \\ \qquad +\,3\eta ^4 \cos (2\phi _2)+12\eta \cos (\phi _1+\phi _2)+ 36\eta \cos (\phi _1) \\ \qquad +\,48\eta \cos (\phi _2)+42\eta ^2+28\eta ^3+5\eta ^4+12\eta \cos (\phi _1+2\phi _2)\\ \qquad \left. +\,18\eta ^2 \cos (\phi _1) +48\eta ^2 \cos (\phi _2)+12\eta ^3 \cos (\phi _2)+29\right) . \end{array} \end{aligned}$$

The minimal value of \(D_\mathrm{{min}}=\frac{4 m_1^2 l^2}{3}\) is obtained in the nonphysical case of \(\eta =0\), for \(\phi _1+\phi _2=\pi \).

Finally, consider the case where all three wheels are slipping. The determinant D is obtained as:

$$\begin{aligned}&D=\frac{m_1^3 l^2}{3}(\eta +2) (8\eta +6\cos (\phi _1+\phi _2)\\&\qquad +\,12\eta c_1+12\eta c_2+12\eta ^2\\&\qquad +\,8\eta ^3+\eta ^4+6\eta ^2 c_1+6\eta ^2 c_2+10). \end{aligned}$$

The minimal value of \(D_\mathrm{{min}}=\frac{8 m_1^3 l^2}{3}\) is obtained in the nonphysical case of \(\eta =0\), for \(\phi _1+\phi _2=\pi \).

Appendix C: Dynamics under torque input near singularity

Consider the three-wheel snake robot as a constrained nonholonomic mechanical system where the torques \({{\varvec{\tau }}}(t)\) are the controlled input, while the shape variables \({\mathbf {q}}_s\) are evolving dynamically. The system’s equations of motion (14) and (15) give rise to a \(8 \times 8\) linear system of the form \({{\mathbf {A}}}{{\mathbf {z}}}= {\mathbf {b}}\) where the vector of unknowns is \({{\mathbf {z}}}=(\ddot{{\mathbf q}}_b,\ddot{{\mathbf q}}_s,{{\varvec{\Lambda }}})\). The determinant of this system can be obtained as:

$$\begin{aligned} \begin{array}{l} D= \frac{1}{9}m_1 l^6 \eta ^2 (16\eta -2\cos (2\phi _1-2\phi _2)+3\eta ^2 \\ \qquad -\,2\eta \cos (2\phi _1)-2\eta \cos (2\phi _2) + 14). \end{array} \end{aligned}$$

It can be shown that the minimal value of D is obtained for \(\phi _1=\phi _2= 0\) as

$$\begin{aligned} D_\mathrm{{min}} = \frac{1}{3}m_1 l^6 \eta ^2 (\eta +2)^2 > 0. \end{aligned}$$

This implies that the dynamical system is never singular, so that \((\ddot{{\mathbf q}}_b,\ddot{{\mathbf q}}_s,{{\varvec{\Lambda }}})\) are always finite and bounded. Nevertheless, if the system is driven into a configuration of kinematic singularity where \(\det ({\mathbf {W}}_b)=0\), the body velocity \({\mathbf {q}}_b\) may still grow unbounded according to (6).

We now examine this case in a numerical simulation under torque input. Physical parameters of the robot are chosen as \(l=h=0.1\) m and \(m_0=m_1=m_2=0.17\) kg. Initial conditions are chosen as \({\mathbf {q}}_b(0)= 0\), \(\phi _1(0)= -\phi _2(0)= 0.25 \pi \) and \(\dot{{\mathbf q}}(0)= 0\). The joint torques are constant: \(\tau _1= \tau _2= 0.05\) N mm. That is, the torques drive the system toward a singular configuration where \(\phi _1=\phi _2\). Simulation results of the joint angles \(\phi _i(t)\) and their velocities \(\pm \, \dot{\phi }_i(t)\) are plotted in Fig. 13a, b, respectively. It can be seen that the system reaches a configuration of kinematic singularity \(\phi _1=\phi _2\) at time \(t_s \approx 0.49\) s, marked as dotted vertical line. Interestingly, at the same time the joints’ angular velocities are driven to satisfy \({\dot{\phi }}_1=-\,{\dot{\phi }}_2\), which is precisely the condition given in (8) for bypassing the kinematic singularity. Figure 13c shows a time plot of the velocity norm \(||\dot{{\mathbf q}}(t)||\), indicating that, indeed, it remains finite and bounded. That is, the system is driven passively toward bypassing the kinematic singularity without divergence.

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Yona, T., Or, Y. The wheeled three-link snake model: singularities in nonholonomic constraints and stick–slip hybrid dynamics induced by Coulomb friction. Nonlinear Dyn 95, 2307–2324 (2019). https://doi.org/10.1007/s11071-018-4693-0

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Keywords

  • Under-actuated robots
  • Nonholonomic systems
  • Robotic locomotion
  • Friction
  • Stick-slip transitions
  • Hybrid dynamics