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Nonlinear Model Predictive Control for Constrained Output Path Following

Timm Faulwasser, Rolf Findeisen

arXiv:1502.02468v1eess.SYmath.OC

TL;DR

The paper addresses constrained path following for nonlinear systems when geometric output paths matter but timing is not prescribed. It proposes predictive control for cases with and without velocity assignments, establishes sufficient convergence conditions, and uses transverse normal forms to simplify terminal-region and end-penalty design. A robotics example illustrates the framework, which handles nonlinear dynamics and state and input constraints while enforcing path convergence.

  • Problem

    Constrained output path following requires steering nonlinear-system outputs along geometric paths under state and input constraints without pre-specified timing, while existing approaches provide limited structural insight.

  • Method

    The paper develops a sampled-data nonlinear model predictive path-following control scheme for path following with and without velocity assignments, using terminal regions, end penalties, and transverse normal forms.

  • Results

    The proposed scheme provides sufficient conditions guaranteeing path convergence and recursive feasibility for constrained output path-following problems.

  • Takeaways & Limitations

    The framework handles nonlinear dynamics, output-space reference paths, and state and input constraints, while transverse normal forms simplify computation of terminal regions and end penalties.

Abstract

from arXiv · show

We consider the tracking of geometric paths in output spaces of nonlinear systems subject to input and state constraints without pre-specified timing requirements. Such problems are commonly referred to as constrained output path-following problems. Specifically, we propose a predictive control approach to constrained path-following problems with and without velocity assignments and provide sufficient convergence conditions based on terminal regions and end penalties. Furthermore, we analyze the geometric nature of constrained output path-following problems and thereby provide insight into the computation of suitable terminal control laws and terminal regions. We draw upon an example from robotics to illustrate our findings.

I. INTRODUCTION

Path-following steers a nonlinear system’s output along a geometric reference without prescribing timing, while prioritizing path deviation and respecting constraints. The paper extends predictive control to constrained output paths and studies convergence and terminal-region design.

  • Path-following motivation: Path following steers a system output along a geometric curve while allowing the controller to adjust speed along the reference.Unlike trajectory tracking, timing is not predefined because velocity is secondary to minimizing geometric deviation.
  • Path-following motivation: Geometric and Lyapunov/backstepping approaches have difficulty directly considering input and state constraints.This motivates nonlinear model predictive control tailored to path-following problems.
  • Output-space formulation: Output-space paths are important for applications including robots, autonomous vehicles, ships, and unmanned aerial vehicles, but earlier predictive approaches often focused on state-space paths.Output path following is therefore needed for realistic movement tasks defined through system outputs.
  • Research gap: Prior convergence results used terminal regions and end penalties, but did not provide insight into the structure of constrained output path-following problems.The paper addresses this structural gap through geometric analysis of nonlinear systems.
  • Contribution: The paper develops a sampled-data NMPC framework for constrained output paths with and without velocity assignments, giving sufficient conditions for convergence and recursive feasibility.It also uses transverse normal forms to support computation of stabilizing terminal regions and end penalties.

III. MODEL PREDICTIVE PATH-FOLLOWING CONTROL

MPFC repeatedly solves an optimal control problem to compute both the physical input and the path timing. Its constraints enforce admissible system behavior and monotone forward motion, while terminal constraints support path-following convergence.

  • MPFC formulation: Model predictive path-following control computes the system input and reference evolution together by repeatedly solving an optimal control problem.The scheme is applied at sampling instances in receding-horizon fashion.
  • MPFC formulation: The OCP optimizes real system inputs and virtual path-parameter inputs over a prediction horizon subject to system and timing-law constraints.The virtual input belongs to a compact set containing zero in its interior.
  • Constraint handling: The timing constraint keeps the path parameter within its prescribed interval and enforces nonnegative path-parameter velocity for monotonous forward motion.Restricting virtual inputs also avoids impulsive solutions of the path-parameter dynamics.
  • Dynamic feedback: At each sampling time, measured plant state initializes the OCP, while the timing-law initialization uses the previous predicted trajectory.Consequently, the path-parameter state is an internal state and MPFC is a dynamic feedback strategy.
  • Terminal design: The terminal constraint places the predicted augmented state in a terminal region, even though the running cost penalizes only outputs and inputs.This state-space terminal condition supports reformulating output path following as manifold stabilization under suitable assumptions.
  • Computational scope: Efficient numerical implementation is outside the paper’s scope, although the OCP is described as a typical NMPC problem with terminal constraints and penalties.Existing real-time-feasible NMPC tools may therefore be applied to the increased state and input dimensions.

B. Sufficient Convergence Conditions

The MPFC scheme achieves constrained output path-following convergence under assumptions on system trajectories, path–state consistency, and the cost function, using a suitable terminal region and terminal penalty.

  • Motivation: Receding-horizon optimal inputs alone do not guarantee closed-loop stability or convergence of the output to the path.The convergence analysis therefore introduces explicit sufficient conditions for MPFC.
  • Assumptions: The assumptions require locally well-behaved system dynamics and absolutely continuous trajectories for admissible inputs.These conditions support existence and uniqueness of solutions and application of Barbalat’s Lemma.
  • Assumptions: The reference path must lie in the interior of the output image of the state constraints, and the cost must lower-bound output-path error through a class K function.The cost explicitly penalizes path-following outputs rather than treating the problem as set-point stabilization.
  • Theorem conditions: A compact terminal region and differentiable positive semi-definite terminal penalty, together with admissible invariant terminal controls, provide the theorem’s convergence conditions.The terminal controls keep predicted trajectories in the terminal region over a nonzero interval, while the initial optimal control problem must be feasible.
  • Convergence result: Under these conditions, MPFC is recursively feasible, satisfies state and forward-motion constraints, and converges to the path and the desired path-parameter target.The proof uses concatenation with terminal controls, value-function decrease, and Barbalat’s Lemma.
  • Scope: The guarantee concerns output convergence rather than Lyapunov-like state stability, so internal states may remain bounded without asymptotically converging.Terminal constraints ensure predicted states remain in the admissible state–path-parameter region in the nominal case.

C. Extension to Predictive Path Following with Velocity Assignment

The MPFC framework extends to velocity-assigned path following by changing the cost and removing boundedness constraints on the path parameter, while retaining terminal-region-based convergence guarantees.

  • Modified formulation: Velocity-assigned path following changes the cost lower bound from path-parameter error to error in the assigned velocity.The modified cost satisfies ψ(∥e, ˙θ−˙θref∥) ≤ F(e, ˙θ, u, v).
  • Modified formulation: Because the path parameter may grow unbounded, the constraint on path-parameter states is dropped by setting Z = R^r.The modified scheme keeps state constraints while allowing unbounded θ.
  • Scope: Sampled-data continuous-time NMPC generally establishes asymptotic convergence rather than Lyapunov stability because open-loop inputs are applied between sampling instants.This limitation is shared with sampled-data NMPC for set-point stabilization.
  • Convergence result: Under these modified conditions, the MPFC scheme solves the velocity-assigned path-following problem.The proof establishes recursive feasibility and constraint satisfaction similarly to Theorem 1, with a modified final convergence argument.

IV. DESIGN OF SUITABLE TERMINAL REGIONS AND END PENALTIES

Designing terminal regions and end penalties is challenging because it generally requires constructing a locally admissible controller for the constrained path-following dynamics.

  • Design challenge: Terminal-region construction generally involves designing a locally admissible controller and a corresponding end penalty.The paper introduces technical results to support this design for MPFC.

A. Trivial End Penalties

The paper shows how invariant terminal controls can generate suitable time-dependent end penalties and, under exponential cost decrease, make a zero end penalty sufficient for convergence.

  • Terminal-region conditions: A compact terminal region rendered controlled positively invariant by terminal controls is the foundation for constructing a valid end penalty.The terminal controls must keep every trajectory starting in the region inside it for all future time.
  • Constructing end penalties: Under the lemma’s conditions, a time-dependent end penalty exists that satisfies Theorem 1’s convergence requirements.The result connects terminal-control behavior directly to the sufficient conditions for MPFC convergence.
  • Constructing end penalties: The constructed penalty can decay exponentially along terminal-control trajectories and satisfy the required cost-decrease condition.The bound has the form ϕE(t) ≤ c e^(-α(t−t0)).
  • Penalty equivalence: Adding a time-only terminal penalty changes the objective by a constant for a fixed sampling instant, so it does not change optimal inputs.The two objective functionals differ only through the terminal penalty’s value at tk + T.
  • Trivial end penalty: A zero terminal penalty combined with a suitable terminal region still yields a convergent MPFC scheme for Problem 1.The equivalence argument transfers Theorem 1’s conclusions from the constructed penalty to E(t, x(t), z(t)) = 0.
  • Design implication: Exponentially cost-decreasing terminal controls remove the need to determine terminal penalties for stability, although penalties may improve closed-loop performance.The claim is framed from the stability perspective of the terminal design.

B. Geometric Structure of Path-following Problems

The paper interprets constrained output path following as stabilization of a path manifold in the augmented state space. A transverse normal form separates path-error directions from unspecified internal directions, clarifying why terminal-region design is difficult.

  • Transverse normal form: Under the vector-relative-degree assumption, the augmented system admits a local transverse normal form with coordinates (ξ, η).The transverse coordinates are built from the path error and its derivatives, while η contains the remaining coordinates.
  • Local structure: The augmented decoupling matrix is upper triangular, with the original system’s decoupling matrix as its upper-left block and full rank locally.This yields the augmented vector relative degree (r1, …, rny, ˆr)ᵀ on a neighborhood of the considered state set.
  • Transverse normal form: The transverse coordinates ξ represent the path error and its derivatives, whereas η describes directions not specified by the normal-form construction.The ξ directions are transverse to the manifold of trajectories traveling along the path.
  • Geometric interpretation: Output path following is locally equivalent to stabilizing a path manifold in the augmented state space, characterized by ξ = 0.The manifold includes states and path-parameter states, and its stabilization underlies path convergence.
  • Terminal design: Computing suitable terminal regions and end penalties is difficult because path-parameter constraints place the terminal path point on the boundary and internal dynamics must be considered.A single-point ellipsoidal terminal region can collapse in path-parameter directions under these constraints.
  • Terminal design: Dropping terminal constraints can make recursive feasibility difficult to guarantee, especially when state constraints and forward motion impose constraints on virtual path-parameter states.The paper notes that removing these constraints may require abandoning state constraints or the forward-motion requirement.

V. EXAMPLE: FULLY ACTUATED ROBOT

The example uses a fully actuated two-degree-of-freedom planar robot whose joint-space path-following dynamics are subject to state and input constraints.

  • Robot model: The example considers a fully actuated planar robot with two degrees of freedom and joint-angle and joint-velocity states.The robot model includes inertia, centrifugal and Coriolis forces, and gravity.
  • Outputs and task: The output is the joint-angle vector, while the Cartesian tool position is an additional output and the inputs are joint torques.The considered path-following task is defined in the joint space.
  • Constraints: The robot is operated under box constraints on its states and inputs.These constraints are incorporated into the constrained path-following setup.

A. Simulation Results

The simulations demonstrate constrained path convergence for the robot, including adjustable reference speed and satisfaction of input constraints. The controller also converges from a range of initial conditions.

  • Controller setup: The MPFC cost penalizes the path error and its time derivative, using Q = diag(10^5, 10^5, 10, 10, 5) and R = diag(10^-3, 10^-3, 10^-4).The terminal penalty is chosen as zero because the constructed terminal region satisfies the stated conditions.
  • Controller setup: The terminal region combines an ellipsoidal restriction on transverse directions with a polyhedral region for the virtual states.The transverse region uses ξᵀPξξ ≤ 3.13, while η is restricted to Eη.
  • Simulation settings: The simulations use V = [−50, 50], a prediction horizon of 0.75s, sampling time δ = 0.005s, and 20 shooting intervals.The optimal control problem is solved repeatedly using direct multiple shooting.
  • Simulation results: The joint positions converge rapidly to the reference, the path parameter reaches θ = z1 = 0, and both input torques satisfy their constraints.The virtual input adjusts the speed along the reference during convergence.
  • Simulation results: The proposed MPFC scheme ensures path convergence for a range of initial conditions and convergence on the path while respecting state and input constraints.The result is illustrated in both joint-angle and Cartesian output spaces.

APPENDIX A COMPUTATION OF A TERMINAL REGION FOR THE EXAMPLE

The appendix constructs a transverse normal-form representation for the robot and uses it to support computation of a terminal region for the augmented system.

  • Robot model: The model components B, C, g, and hca define inertia, velocity-dependent forces, gravity, and Cartesian tool position for the robot.Their parameter expressions and system parameters are provided in the appendix and Table I.
  • Relative degree: The robot has global vector relative degree r = (2, 2)ᵀ with respect to the joint-angle output.An integrator chain of length two is therefore used for the path-parameter dynamics.
  • Coordinate transformation: A coordinate transformation Φ maps the augmented robot dynamics into transverse normal form.The inverse transformation expresses joint angles using transverse coordinates and the path parameter.
  • Coordinate transformation: Only the virtual states appear in η because the robot has no internal dynamics relative to its joint-angle output.The robot state dimension equals the vector relative degree sum.
  • Coordinate transformation: For the chosen path parametrization, Φ is a global diffeomorphism, making the transformed representation globally valid for the example.The transformed dynamics are then obtained directly from Φ.

C. Design of a Terminal Region

The terminal-region design separates the path-parameter dynamics from the transverse dynamics, using invariant regions and feedback laws that respect constraints and promote convergence. The resulting regions combine a polytopic set for the path parameter with an ellipsoidal Lyapunov level set for transverse directions.

  • Terminal-region construction: Terminal control is designed first for the path-parameter dynamics and then for the transverse dynamics in augmented transverse normal form.The path-parameter dynamics are decoupled from the other states, enabling this sequential construction.
  • Path-parameter region: The path-parameter dynamics form a double integrator, so a positively invariant polytopic region is constructed under linear feedback while preserving the state constraint polytope.Ellipsoidal regions would shrink to a point because the origin lies on the boundary of the constraint set.
  • Path-parameter region: Negative feedback gains enforce convergence of the path parameter, while an additional gain condition makes boundary velocity vectors point toward the path-parameter axis.The gain restrictions also place the relevant eigenspaces in specified quadrants and avoid oscillations.
  • Transverse region: The transverse feedback achieves global transverse feedback linearization, and its linear closed-loop dynamics are stabilized using the matrix Kξ and a corresponding Lyapunov function.The feedback includes a stabilizing state-feedback term and a feedforward term based on the path curvature dynamics.
  • Transverse region: The transverse terminal region is chosen as a Lyapunov-function level set whose size is constrained by bounds on model terms, path derivatives, and tightened input and state constraints.Computing the largest admissible level set is formulated as a simplified convex maximum-volume ellipsoid problem when the relevant bounds are positive.
  • Robot example: For the robot example, the path-parameter region uses η2 ∈[0, 0.4] and η1 ∈[−5.3, 0], while the transverse ellipsoidal level set is obtained numerically.The reported numerical construction uses Kη = (−0.1, −1.33), an LQR-based Kξ, and γ = 1.77.

D. Derivation of a Terminal Penalty

The terminal penalty is derived by evaluating the quadratic running cost under the terminal feedback inside the constructed terminal region. Exponential convergence of the transverse and path-parameter states supports the required terminal-cost conditions.

  • Penalty construction: The derivation begins by requiring an end penalty that satisfies the conditions of Theorem 1 or Proposition 1.
  • Penalty construction: The MPFC running cost is quadratic and is expressed in transverse coordinates using the transverse state, path parameter, and control inputs.
  • Convergence: Inside the terminal region, the terminal feedback yields exponential convergence of the transverse state ξ, path-parameter state η, and virtual input vE.The associated convergence bounds are finite for all initial conditions in the terminal region.
  • Convergence: Exponential convergence of η implies exponential convergence of the path position p(η1), the state x1, and the nonlinear term g(x1) toward their endpoint values.Because the cost is quadratic, these convergence properties provide the estimates needed for the terminal penalty conditions.
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