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Safety-Critical Kinematic Control of Robotic Systems
Andrew Singletary, Shishir Kolathaya, Aaron D. Ames
TL;DR
Robotic kinematic controllers are useful, but safety methods can rely heavily on uncertain system models and may not account for velocity-based low-level tracking. This paper develops kinematic and energy-based CBF controllers, extending the latter to underactuated systems. It establishes forward-invariance guarantees under stated conditions, including for an underactuated case.
Problem
Existing safety constraints can depend heavily on the system model, while users may have only velocity inputs and limited knowledge of the dynamics.
Source: §I · p. 1Method
The paper extends CBFs to kinematic control and adds kinetic energy to a dynamically consistent safe-set extension, with a construction for underactuated systems.
Source: §IV · p. 3, §V · p. 5Results
The energy-based constraint is a CBF that guarantees forward invariance of the dynamically extended safe set, and the underactuated construction guarantees safety of its corresponding set.
Source: §IV · pp. 4–5, §V · p. 6Takeaways & Limitations
The dynamics-level controller’s barrier derivative depends on kinematics, gravity, and inertia, rather than requiring the Coriolis matrix.
Source: §IV · p. 4Kvel may be unknown for industrial actuators and can typically be determined from experimental data.
Source: §IV · p. 5
Abstract
from arXiv · showhide
Over the decades, kinematic controllers have proven to be practically useful for applications like set-point and trajectory tracking in robotic systems. To this end, we formulate a novel safety-critical paradigm for kinematic control in this paper. In particular, we extend the methodology of control barrier functions (CBFs) to kinematic equations governing robotic systems. We demonstrate a purely kinematic implementation of a velocity-based CBF, and subsequently introduce a formulation that guarantees safety at the level of dynamics. This is achieved through a new form CBFs that incorporate kinetic energy with the classical forms, thereby minimizing model dependence and conservativeness. The approach is then extended to underactuated systems. This method and the purely kinematic implementation are demonstrated in simulation on two robotic platforms: a 6-DOF robotic manipulator, and a cart-pole system.
I. INTRODUCTION
Kinematic control is useful for robotic behaviors, but safety methods face high model dependence and often lack a general way to incorporate velocity-based low-level tracking. The paper introduces an alternative energy-based CBF formulation and extends the approach to velocity inputs and underactuated systems.
I. INTRODUCTION
- Existing safeguarding methods depend heavily on system models, while users of many robotic platforms can provide only desired velocity inputs and may not know the dynamics.The introduction identifies this mismatch as a limitation for safely incorporating low-level velocity tracking into high-level control.Source: §I · p. 1
- The paper introduces an energy-based zeroing CBF formulation that is well defined on the safe-set boundary and exterior while minimizing added conservatism.The approach is used to provide guarantees for velocity control inputs with partial model information and is extended to underactuated systems.Source: §I · p. 1
- The method is demonstrated in simulation on a 6-DOF manipulator and a cart-pole system under different levels of uncertainty.The paper also compares its dynamics-level formulation with the purely kinematic case on the manipulator.Source: §I · p. 1
II. BACKGROUND
The background defines safety as forward invariance of a set and presents control barrier functions as a way to guarantee it. It also describes a quadratic-program safety filter that minimally modifies a desired input.
II. BACKGROUND
- Safety means that every trajectory starting in the safe set remains there for all future time.The background defines safety as forward invariance under a feedback control law.Source: §II · p. 2
- A control barrier function on the safe set makes a controller satisfying the barrier inequality render that set safe.When the function is a CBF on the larger domain, the set is also asymptotically stable there.Source: §II · p. 2
- The CBF safety filter uses a quadratic program to minimally modify the desired control input while enforcing safety.For unconstrained inputs, the background gives an explicit solution based on whether the desired input already satisfies the barrier condition.Source: §II · p. 2
III. SAFETY-CRITICAL KINEMATIC CONTROL
The section develops velocity-based kinematic safety control and tests it on manipulator obstacle avoidance. In that example, avoidance depends on the barrier parameter, and the method does not guarantee safety of the full dynamics.
III. SAFETY-CRITICAL KINEMATIC CONTROL
- The kinematic mapping x=y(q) relates system configuration q to task or output variables x, with k≥n degrees of freedom assumed.The controller treats joint velocity as its input and can use a pseudoinverse-based law for trajectory tracking with exponentially stable error dynamics.Source: §III · p. 2
- The velocity-based safety filter keeps the desired joint velocity when safe and otherwise intervenes to enforce the barrier condition.The section presents a closed-form controller for this intervention.Source: §III · p. 3
- The manipulator obstacle-avoidance example defines safety as keeping the end-effector at least distance d from an obstacle.Its obstacle-clearance constraint is expressed in joint configuration using the end-effector position and manipulator Jacobian.Source: §III · p. 3
- For α in [0.5, 1], the manipulator avoids the obstacle; for α in [2, 3], it does not.The velocity-based CBF does not account for system dynamics or low-level tracking ability, so safety is not guaranteed, though it is achieved for the stated parameter range.Source: §III · p. 3, Figure 2 · p. 3
IV. FROM KINEMATICS TO DYNAMICS
The section extends kinematic safety constraints with kinetic energy to form a zeroing CBF for robotic dynamics, using limited model information. It also applies the construction to desired-velocity tracking and demonstrates a safety–performance trade-off in simulation.
IV. FROM KINEMATICS TO DYNAMICS
- The construction extends a kinematic safe set using kinetic energy; its zeroing CBF is well-defined on the set boundary and outside it, unlike the reciprocal formulation.The paper relates the energy-based safe set to the original safe set as αe increases.Source: §IV · p. 3
- For fully actuated systems with invertible B, the energy-based CBF guarantees forward invariance of SD using kinematics, gravity, and inertia, without requiring the Coriolis matrix.The section defers the underactuated case to Section V.Source: §IV · pp. 3–4
- The energy-based constraint is a CBF without requiring the kinematic constraint to have relative degree one, except on the boundary of the original safe set.Source: §IV · p. 4
- Under stated inertia and gravity bounds, the corollary permits replacing the safety-critical QP constraint while retaining safety of SD.Source: §IV · p. 4
- Safety holds across αe values in the manipulator simulation, while larger αe lets the robot move faster and closer to obstacles.The example uses cu = 5λmax(D).Source: Figure 3 · p. 5, §IV · p. 5
- For desired-trajectory tracking through velocity commands, Theorem 3's QP guarantees forward invariance of SD; an unknown Kvel can typically be estimated experimentally or addressed with adaptive CBFs.Source: §IV · p. 5
V. UNDERACTUATED SYSTEMS
For underactuated systems, the method transforms coordinates to incorporate the safety constraint, then constructs a dynamically consistent barrier and controller. The resulting controller guarantees forward invariance of a certified safe set contained within the original safe set.
V. UNDERACTUATED SYSTEMS
- The method treats h(q) as a coordinate by choosing Φ(q) = (w(q), h(q)) to be a diffeomorphism.This transforms the dynamics so the safety constraint can be handled in the underactuated formulation.Source: §V · p. 5
- Eliminating the transformed dynamics’ ¨w term yields a Schur-complement term D_h, which is symmetric positive definite.The formulation also introduces transformed terms C_h and G_h.Source: §V · p. 6
- The formulation assumes h is inertially coupled to the input; if this fails in part of Q, the method restricts the domain to a subset Q_u.For the cart-pole, pole angle is not inertially coupled to the input when the pole is horizontal.Source: §V · p. 6
- The underactuated controller guarantees forward invariance of the certified set bSD, which is contained within the original safe set S.The guarantee holds for states in bSD.Source: §V · p. 6
- The paper also gives a robust-QP constraint using bounds on transformed dynamics terms, but those bounds may not hold throughout Q × R^k.Their validity depends on the coordinate transformation Φ.Source: §V · p. 6
- The cart-pole example uses cart position x as w and constrains pole angle θ to [5π/6, 7π/6].The example applies QP (28); Fig. 4 is captioned as a cart-pole system with an energy-based CBF.Source: §V · p. 6, Figure 4 · p. 6