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Kinodynamic Motion Planning: A Novel Type Of Nonlinear, Passive Damping Forces And Advantages

Ahmad A. Masoud

arXiv:1606.09270v1cs.ROeess.SY

TL;DR

The paper addresses the difficulty of turning harmonic-potential guidance into dynamic robot control without losing convergence, speed, or applicability beyond dissipative systems. It augments the gradient field with nonlinear anisotropic damping forces, showing theoretically and through simulations that the resulting signal suppresses inertial transients while preserving guidance and handling external forces without full dynamics knowledge.

  • Problem

    Converting harmonic-potential guidance into a control signal can produce inertia-related transients, speed-bound restrictions, and limited treatment of external forces.

  • Method

    The approach augments the harmonic potential gradient with nonlinear anisotropic damping forces that attenuate disruptive motion while retaining guidance-aligned motion.

  • Results

    The method provides a well-behaved control signal, suppresses inertia-induced artifacts, and supports dissipative and externally forced systems without exact system dynamics.

  • Takeaways & Limitations

    NADF-based control extends harmonic potential fields from kinematic guidance toward provably correct kinodynamic planning with agile response.

Abstract

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This article extends the capabilities of the harmonic potential field approach to planning to cover both the kinematic and dynamic aspects of a robot motion. The suggested approach converts the gradient guidance field from a harmonic potential to a control signal by augmenting it with a novel type of damping forces called nonlinear, anisotropic, damping forces. The combination of the two provides a signal that can both guide a robot and effectively manage its dynamics. The kinodynamic planning signal inherits the guidance capabilities of the harmonic gradient field. It can also be easily configured to efficiently suppress the inertia-induced transients in the robot trajectory without compromising the speed of operation. The approach works with dissipative systems as well as systems acted on by external forces without needing the full knowledge of the system dynamics. Theoretical developments and simulation results are provided in this article.

I. Introduction

Harmonic potential fields offer fast, provably correct guidance, but conventional planning methods face realistic-setting performance and kinodynamic limitations. The paper proposes converting harmonic guidance directly into navigation control using nonlinear anisotropic damping forces.

  • Potential-field planning can guide autonomous agents toward reachable goals and indicate when tasks are intractable.
  • Navigation control directly converts environmental data, robot goals, and behavioral constraints into a control signal, while retaining potential fields’ high response speed.
  • Earlier gradient-control combinations require initial speed below an upper bound, provide limited transient management, and are not described for systems affected by external forces.
  • Kinematic guidance must otherwise be converted into a control signal by a lower-level controller, as in sliding-mode approaches that do not require full system dynamics.
  • The proposed method augments harmonic guidance with nonlinear anisotropic damping to suppress inertia-induced artifacts while preserving agile response and handling dissipative or externally forced systems.
  • The paper develops the approach through background, NADF theory, applications to dissipative and externally forced systems, simulations, and conclusions.

II. Background

Harmonic potential fields address local minima by imposing Laplace-equation structure and boundary conditions, producing a guidance trajectory. That trajectory is only a reference, and naive damping can still cause collisions.

  • Harmonic potential fields eliminate the local-minima problem by satisfying the Laplace equation inside the workspace and constraining potential properties at its boundary.
  • The harmonic gradient field generates a trajectory guaranteed to remain in the workspace and converge to the target.
  • Harmonic potential fields are a special case of PDE-ODE motion planners defined through a boundary-value problem and a nonlinear trajectory-generating system.
  • The generated trajectory is a reference for a lower-level controller rather than the robot’s direct control signal.

III. The NADF Approach

The NADF approach converts harmonic guidance into navigation control by damping only motion orthogonal to the guidance direction. Propositions establish convergence and arbitrarily small deviation from the kinematic trajectory under stated system assumptions.

  • Linear damping treats useful guidance-aligned motion and unwanted inertia-induced motion equally, motivating direction-selective attenuation.
  • NADFs use the gradient as both guidance and dynamic actuation, leaving guidance-conforming motion unaffected while attenuating the orthogonal disruptive component.
  • For fully actuated second-order dissipative systems, the harmonic gradient combined with NADF guarantees global asymptotic convergence.
  • The equilibrium analysis uses the harmonic potential’s unique minimum at the target and nonsingular Hessians at other critical points to identify the limiting invariant set.
  • A suitable damping coefficient can make the maximum deviation between dynamic and kinematic trajectories arbitrarily small, preserving spatial constraints.
  • The construction assumes bounded speed, bounded Coriolis-related terms, and a positive-definite bounded inertia matrix.
  • NADF can be made arbitrarily large without slowing the system because it is zero when motion follows the guidance field.

V. Systems with External Forces

For systems with external forces, the NADF controller combines harmonic guidance, damping, and force compensation to keep dynamic motion near the target and kinematic path. A clamping controller and iterative error cancellation further address residual steady-state error while preserving stability.

  • External-force systems: The NADF approach extends constrained motion control to mechanical systems subject to external forces such as gravity.The system includes inertia, centripetal, Coriolis, gyroscopic, external, and applied control forces.
  • External-force systems: A gradient field combined with sufficiently strong NADF can make the dynamic trajectory closely follow the kinematic trajectory from the initial state to the target.The approach is described for systems whose dynamics include external forces.
  • Error compensation: The proposed alternative to an integrator preserves stability and can cancel external-force error, bringing the dynamic trajectory arbitrarily close to the target.This approach gives up the controller's ability to drive motion arbitrarily close to the target without the additional error-cancellation procedure.
  • Clamping control: Clamping control acts only within a neighborhood of the target, remaining inactive during target-directed motion and reactivating when motion moves away.Its effect is localized to a hyperspace of radius σ around the target.
  • Clamping control: For any positive Kc, clamping control keeps the mechanical system stable while limiting steady-state error below a desired level.The controller can ensure lim t→∞ ||x(t)−xT|| ≤ ε < σ under the stated positive-gain conditions.
  • Iterative error cancellation: A fixed-point iteration uses successive settling states and converges to the target, with the target acting as a stable attractor fixed point.The construction uses control input and robot-coordinate velocity to assess convergence, and proves x_i = xT at the fixed point.

VI. Results

Simulations examine NADF motion in cluttered environments, speed-limited motion, and settling-time behavior. The results indicate that NADF suppresses nonconvergent motion components while retaining fast convergence and avoiding the usual settling-time/path-deviation compromise.

  • Speed-limited motion: Without a speed limit, radial speed continuously increases before the target because NADF applies no friction along the radial trajectory component.The result motivates imposing an upper speed bound when fast response must be combined with control over excessive motion.
  • Speed-limited motion: A speed limit prevents uncontrolled radial acceleration, but increases settling time from 12 seconds to 16 seconds while reducing control effort.The imposed speed limit is 5 m/sec.
  • Settling-time comparison: Linear viscous damping has a convex settling-time relationship with B, with low damping causing oscillation and high damping delaying response.The minimum occurs at one value of the linear viscous friction coefficient.
  • Settling-time comparison: NADF settling time decreases rapidly and strictly as Bd increases because NADF attenuates tangent motion while leaving motion along the gradient field unaffected.Tangent motion delays reaching the target but does not contribute to convergence.
  • Settling-time comparison: Unlike linear viscous damping, NADF avoids a settling-time/path-deviation compromise because both settling time and maximum spatial deviation decrease with Bd.This behavior supports tuning the controller to satisfy both convergence and path-deviation requirements.

4. Point mass with external forces

The NADF approach controls a point mass under external forces using a gradient field augmented with damping and, when needed, clamping. Simulations examine convergence, oscillation suppression, noise robustness, braking, and inertial effects.

  • External-force control: A gradient field combined with NADF and clamping controlled a point mass with constant external forces.The approach was applied to a second-order point-mass system experiencing external forces.
  • External-force control: A sufficiently high Bd drove the mass to the target while avoiding obstacles, although external forces caused drift after target arrival.The controller succeeded in reaching the target and avoiding obstacles, but drift occurred once the target was reached.
  • External-force control: Clamping held the trajectory near the target using only a loose upper-bound estimate of the drift, with overdamped settling and no oscillations.The reported simulation used K=1, Bd=10, and KC=10.
  • Control comparison: Compared with NADF and clamping, sliding-mode control produced a shakier trajectory with oscillations near the target and different control-signal quality and magnitude.The sliding-mode comparison was configured for a 6 sec settling time and 100 N maximum control effort.
  • Robustness and response: Sensor noise had little effect on the NADF trajectory, which maintained a steady path and unchanged travel distance.The test added uniformly distributed wideband noise between (-0.5, 0.5) to wall-position readings with Bd=30.
  • Robustness and response: NADF handled emergency braking and multiple obstructions, while adding mass to a two-robot position-exchange scenario caused the planner to fail.The braking simulation placed a barrier in the robot’s path; the mass-effect comparison contrasted massless robots with robots of 1Kg each.
  • Robustness and response: NADF reduced inertial transients without the severe speed loss caused by linear damping: robot-1 reached its target in two seconds versus about 13 seconds.The comparison used linear damping B=1 and NADF with Bd=10.
  • Robustness and response: In the pendulum test, switching eliminated steady-state error caused by weight across threshold settings, and earlier switching could reduce settling time.The simulations reported little sensitivity to transients and faster settling when switching occurred before complete settling.

VII. Conclusions

The paper extends harmonic potential fields to kinodynamic planning with nonlinear, anisotropic damping forces. It reports a provably-correct, tunable, noise-resistant approach that handles dissipative systems and external forces without exact system dynamics.

  • Conclusions: NADFs extend harmonic potential fields from guidance toward kinodynamic planning by accounting for both motion actuation and guidance.The paper identifies the dual role of the potential-field gradient as both a motion actuator and a guidance provider.
  • Conclusions: The approach is described as provably-correct, easy to tune, flexible, and capable of generating a well-behaved control signal.These properties are stated as conclusions of the paper.
  • Conclusions: The method is reported to resist sensor noise and operate without exact knowledge of system dynamics.A loose upper-bound estimate is stated as sufficient for constructing a well-behaved control signal.
  • Conclusions: The approach applies to dissipative systems and systems influenced by external forces.The conclusion explicitly includes both system classes.

Appendix

The appendix establishes that harmonic functions are Morse functions: their critical points inside the workspace have nonsingular Hessians. Harmonic potential boundary conditions also support navigation-function properties.

  • Proposition and proof: A harmonic function on an open subset of R^N has a nonsingular Hessian at every critical point and is therefore Morse.The proof uses the absence of interior extrema and the fact that a harmonic function cannot be constant on an open subset without being constant throughout the domain.
  • Proposition and proof: Any interior critical point of a harmonic function must be a saddle point because harmonic functions have no local or global extrema inside the open set.Extrema can occur only on the boundary of the domain.
  • Proposition and proof: The proof expands the potential near a critical point with a second-order Taylor series and analyzes its Hessian through eigenvalue decomposition.The eigenvectors form an orthonormal matrix, and the Hessian eigenvalues must be nonzero.
  • Navigation functions: The navigation function in [13] is presented as a special case of a harmonic potential field.Its stated requirements include smoothness, one minimum at the target, the Morse property, and a maximal constant boundary value.
  • Navigation functions: In harmonic navigation, boundary conditions treat the obstacle boundary and target as the workspace boundary and force the minimum to occur at the target.Dirichlet conditions make the potential maximal and constant on the boundary.
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