Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: (a) Chaplygin sleigh with a balanced rotor. The center of mass of the arrangement is at a distance b from the rear contact. The picture on the right shows an illustration of a physical cart to realize the Chaplygin sleigh with castors at the front.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Phase space plot of the Chaplygin sleigh for parameter values m = 1, b = 1, and I = 1. The energies for the trajectories from the innermost trajectory outward are KE = [0.125, 0.5, 1.125, 2].

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Cart parameters for initial conditions of ux(0)=5 and ω(0)=2. Required friction for slip is assumed to be infinite.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Phase space portrait of the reduced equations of the Chaplygin sleigh. The black curves (the rectangular hyperbolas) denote the critical friction, Fc. The shaded region is the set of initial conditions, which leads to pure nonholonomic motion.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: S1→S2→S4 transition from stick to slip and then back to stick mode of motion. Initial conditions are ux(0)=0 and ω(0)=2 (starting in S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively. In (e), the black line is the total kinetic energy and the dashed curve is the translational kinetic energy.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: S1→S2→S1→S2→S4 transition. Initial conditions are (ω(0)=3.3,ux(0)=0)∈S1). (a) The green (in plane trajectory) and red (out of the plane trajectory) represent the stick and slip motion respectively. (b) The dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the blue dashed–dotted curve represents the normal velocity of point P. (c) and (d) show ω(t) and ux(t), respectively.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Trajectories of the cart for (ux(0),ω(0))=(0,5.6) (shown purple) and (ux(0),ω(0))=(0,5.7) (shown red). In both cases, the no-slip trajectory confined to the plane uy = 0 is shown in green.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Number of slip phases against ω(0) for ux(0)=0

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Normalized change in kinetic energy for a large range of initial conditions, (ux(0),ω(0))

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Plot showing the zero contour of the change in kinetic energy surface. Green (shaded) region represents positive change, and white (unshaded) region represents negative change.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: (a) Values of (ux(0),Δω) for which Δθ is positive (blue squares) and negative (red asterisks), respectively. The ΔKE=0 curve is also pictured in black. (b) The intersection of the ΔKE=0 curve with the Δθ=0 curve is shown. Blue (region 4) marks initial conditions where 10−5<ΔKE<5(10−5). Similarly, cyan (region 5) corresponds to |ΔKE|<10−5 and for initial conditions marked green (region 6) −5(10−5)<ΔKE<−10−5. For red (region 1) initial conditions 5(10−5)<Δθ<5(10−4), for yellow (region 3) |Δθ|<5(10−5), and for purple (region 2) −5(10−4)<Δθ<−5(10−5).

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Parallel translation for the case where ΔKE=0 and Δθ=0. Initial conditions are ux(0)=1.539646 and Δω=3.020854. In (c), the dashed green curve represents the value of friction necessary to stop the slipping of the cart, the solid magenta line is the actual friction, and the dashed–dotted curve represents thenormal velocity of point P. In (f), the black line represents the total kinetic energy, and the blue-dashed curve represents the translational kinetic energy.

Date of download: 10/25/2017 Copyright © ASME. All rights reserved. From: Stick–Slip Motion of the Chaplygin Sleigh With a Piecewise-Smooth Nonholonomic Constraint J. Comput. Nonlinear Dynam. 2017;12(3):031021-031021-8. doi:10.1115/1.4035407 Figure Legend: Perturbation is applied periodically such that the cart avoids a rectangular obstacle. In (a), the path of the cart is shown, and ω is shown in (b).