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Real-Time Reference Shaping for Servo Systems

Zehui Lu, Yebin Wang, Go Sato, Fujita Tomoya

arXiv:2608.30825v1eess.SY

TL;DR

The paper addresses the computational difficulty of real-time motion planning under nonlinear, state-dependent actuator constraints. It develops an analytically characterized reference reshaper and a trajectory compensator, achieving large speedups while maintaining feasibility and enabling operation near 100 kHz.

  • Problem

    Real-time servo planning must satisfy actuator constraints, but optimal-control methods are expensive and simplified profiles can produce infeasible references or require conservative limits.

  • Method

    The method uses KKT-based analytical characterization, finite candidate enumeration through closed-form expressions and eigenvalue computation, and online trajectory compensation for position mismatch.

  • Results

    The algorithm recovers the exact optimal solution and achieves 337× speedup over prior reference reshaping and 1067× over numerical optimization, enabling frequencies approaching 100 kHz.

  • Takeaways & Limitations

    Reference reshaping and compensation provide a unified real-time procedure that maintains actuator feasibility and accurate terminal position tracking without additional trajectory optimization.

Abstract

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This paper addresses real-time motion planning for servo systems subject to nonlinear, state-dependent actuator constraints. A reference reshaping method is proposed that combines analytical optimality with high computational efficiency. Using Karush-Kuhn-Tucker conditions, the problem structure is explicitly characterized, and it is shown that the optimal solution lies within a finite set of candidate points. The complete solution set is constructed via closed-form expressions and a small-scale eigenvalue problem, yielding a deterministic algorithm that recovers the exact optimal solution without iterative optimization or root-finding. To address kinematic mismatch induced by aggressive commands, a real-time trajectory compensator is introduced to correct accumulated position error while preserving feasibility. Simulation results demonstrate significant computational speed improvements over existing methods, enabling real-time implementation at frequencies approaching 100 kHz.

I. Introduction

Servo motion planning must balance fast point-to-point movement with actuator-level feasibility. The paper develops deterministic reference reshaping and compensation methods to preserve feasibility, correct position mismatch, and support high-frequency implementation.

  • System context: Servo systems use trajectory generation followed by motor tracking control for high-speed point-to-point positioning.The architecture is used in manufacturing systems including CNC machines, robotic manipulators, and lithography systems.
  • Motivation: Computationally expensive optimal-control methods are difficult to deploy in real-time industrial systems, while simplified profiles may violate torque, current, or voltage limits.Conservative planner limits can avoid violations but sacrifice productivity.
  • Analytical reference reshaping: The proposed reference reshaper characterizes constraint boundaries analytically and enumerates finite candidate solutions using closed-form expressions and a 6 × 6 eigenvalue computation.This avoids iterative root-finding and initialization-dependent procedures.
  • Computational performance: The deterministic algorithm recovers the exact optimal solution while achieving 337× speedup over prior reference reshaping and 1067× over numerical optimization.The speed improvements enable real-time operation at frequencies approaching 100 kHz.
  • Trajectory compensation: A trajectory compensator corrects accumulated position error from aggressive commands by inserting a constant-velocity phase without violating actuator constraints.The compensator exploits the symmetry of time-optimal trajectories.
  • Integrated architecture: The reshaper operates between motion planning and motor control, adjusting desired acceleration using the current state to satisfy actuator torque-speed constraints.The system is modeled around an SPMSM driving a one-dimensional work-tool axis.

III. Preliminaries

This section models motor torque capacity as a speed-dependent analytical bound and summarizes its key properties. It also introduces analytical conversion from desired torque to motor currents.

  • Motor torque capacity: The motor torque bound is defined by piecewise analytic functions of speed under maximum-torque-per-ampere control.The provided example applies when Φpm/Ld − Imax < 0.
  • Torque-bound properties: Figure 2 marks ωr(β) and ωs(β) and associates the three torque-bound segments with distinct plotted lines.The vertical markers identify the transition speeds between the piecewise regions.
  • Torque-bound properties: The torque is constant up to ωr(β), decreases monotonically toward the maximum speed ωm(β), and is symmetric for positive and negative speed.At ω = ωm, the torque is zero; when Φpm/Ld − Imax ≤ 0, ωm(β) = ∞.
  • Motor torque capacity: The torque-bound plot distinguishes feasible and infeasible operating regions for the case Φpm/Ld < Imax.The figure visualizes the motor torque limit across operating conditions.
  • Torque-to-current conversion: Given motor speed, desired torque, and the analytical bound, the desired motor currents id,des and iq,des can be computed analytically.This procedure is referred to as “Torque-Speed To Current” in Fig. 1.

B. Existing Reference Shaping Algorithm

The existing reference reshaping algorithm solves a constrained optimization at each interval by selecting the feasible acceleration closest to the planner’s desired acceleration. KKT conditions reduce the search to a finite set of speed candidates consisting of stationary points and feasible-region boundaries.

  • Optimization formulation: At each timestamp, the reference reshaper receives the planner’s desired acceleration and solves a constrained optimization over the next time interval.The interval has duration ∆, with ades,k+1 defined for [tk, tk+1).
  • Optimization formulation: The torque margin γ ∈ (0, 1] permits slight transient overshoot during trajectory tracking.It is prescribed near 1.
  • Finite candidate search: KKT conditions show that the optimal motor speed lies among at most six critical points rather than requiring a search over the full feasible set.This contrasts with classic reference or command governors that solve a constrained optimization for every reference point.
  • Finite candidate search: Candidate optimal speeds comprise stationary points of the objective and feasible-region boundary points.Boundary candidates belong to ΩL ∪ ΩU ∪ {±ωm(β)}.
  • Geometric interpretation: The optimal acceleration is the feasible point on the kinematic line that has minimal distance to the desired acceleration.Figure 3 depicts the line segment intersecting the positive and negative torque bounds.

IV. Real-Time Reference Reshaping and Trajectory Compensation

The improved reference reshaping algorithm removes nonlinear root-finding from the prior approach by characterizing feasible-boundary roots and searching a finite candidate set. Its real-time implementation is proven optimal, while trajectory compensation addresses position deviation left by reshaping.

  • Improved Reference Reshaping: The algorithm computes feasible constraint-boundary roots using closed-form expressions, a companion-matrix eigenvalue calculation, and branch-specific feasibility checks.The relevant polynomial is sixth-degree, so the method uses Lemma 2 to calculate the lower and upper bounds efficiently.
  • Improved Reference Reshaping: Algorithm 2 eliminates the original computational bottleneck by incorporating the lower-and-upper-bound procedure into a real-time reference reshaper.The method systematically selects the physically admissible solution using the current motor speed.
  • Optimality: Theorem 1 proves that searching the finite solution-candidate set returns the optimal solution of optimization problem (4).This establishes exact optimality for the returned reshaped command.
  • Motivation: Iterative root-finding can converge to only one root and therefore cannot reliably identify the physically admissible solution when multiple feasible roots exist.The proposed method instead explicitly characterizes all real roots before selecting among them.
  • Trajectory Compensation: Dynamic feasibility alone does not account for position deviation caused by modifying the desired acceleration profile, motivating a separate compensation mechanism.The limitation is addressed in the subsequent trajectory-compensation section.

B. Trajectory Compensation and Integrated Algorithm

The trajectory compensator integrates with reference reshaping to correct position mismatch online. It detects the acceleration-switching condition, inserts a symmetric constant-velocity phase, and then resumes the nominal trajectory while preserving efficient real-time operation.

  • Trajectory Compensation: The compensator uses the symmetry of time-optimal bang-bang trajectories to insert a feasibility-preserving constant-velocity phase.The acceleration profile has a single switching point, creating an opportunity for online compensation.
  • Activation: Compensation activates when the desired acceleration changes sign, the reshaped acceleration differs from it, and no prior compensation is active.The algorithm records the velocity at activation and sets the compensation flag.
  • Compensation Phase: During compensation, acceleration is set to zero and motion continues at the recorded constant velocity while compensation time accumulates.The position is updated according to this constant-velocity segment.
  • Compensation Duration: The compensation duration is determined symmetrically: the half-duration is recorded at midpoint position 0.5pT, and the phase ends after 2tcomp,h.Time mapping keeps reshaping aligned with the original trajectory after the inserted segment.
  • Integrated Algorithm: The integrated procedure achieves accurate terminal-position tracking without additional trajectory optimization while retaining high-frequency computational efficiency.Algorithm 3 combines reference reshaping and trajectory compensation in one real-time process.

V. Simulation Results

The simulation section specifies the servo-system parameters used to evaluate dynamic feasibility and computational efficiency.

  • Simulation Setup: The simulations use p = 4, ∆ = 1 ms, γ = 0.97, R = 0.08 Ω, J = 0.15 kg·m2, η = 0.0075, Imax = 40 A, and Vmax = 100 √.These parameters define the evaluated system and controller settings.

3 V. The system

The motor dynamics are propagated using Euler integration with a 0.1 ms time step, and the true motor torque bound is shown in Fig. 2.

  • System Implementation: Euler integration propagates the dynamics with time step ∆s = 0.1 ms, while Fig. 2 displays the true motor torque bound.The numerical propagation step and torque-bound visualization define part of the implementation setup.

A. Position & Current Tracking Controller

The controller architecture uses a 1 kHz position loop to compute desired motor torque from tracking error, followed by a 10 kHz current loop that regulates motor currents through voltage inputs.

  • A. Position & Current Tracking Controller: The position tracking controller operates at 1 kHz and computes desired motor torque from tracking error.
  • A. Position & Current Tracking Controller: The current controller operates at 10 kHz and regulates d- and q-axis current tracking errors through voltage inputs.

B. Reference Reshaper

The reference reshaper is evaluated for dynamic feasibility and computational efficiency, reshaping aggressive commands into feasible trajectories while targeting substantial speed improvements over existing approaches.

  • B. Reference Reshaper: The evaluation measures dynamic feasibility and computational efficiency across baseline, original reshaper, numerical optimization, and proposed reshaper methods.The compared methods are B, A, C, and P.
  • B. Reference Reshaper: The aggressive bang-bang trajectory is dynamically infeasible because of torque-speed characteristics, and reducing maximum acceleration increases motion time.Method B produces torque, current, and voltage constraint violations, including a torque spike around 2.3 s.
  • B. Reference Reshaper: The proposed method reshapes the aggressive trajectory into a feasible one whose actual torque closely tracks the reshaped reference.
  • B. Reference Reshaper: Inductance and permanent magnetic flux affect the analytical reshaping boundaries alongside real-time parameter estimation.
  • B. Reference Reshaper: 337× and 1064× speed improvements over existing methods enable dynamically feasible implementation at frequencies approaching 100 kHz.The framework uses KKT conditions and deterministic evaluation to replace iterative optimization.

Appendix

The appendix analyzes nonlinear boundary equations branch by branch, deriving feasible root conditions and closed-form or eigenvalue-based procedures for constructing admissible solutions.

  • Appendix: The analysis begins with the root-finding problem for the upper-bound equation and treats the lower-bound derivation analogously.
  • Appendix: For the first branch, the upper and lower feasible solution intervals depend on the sign of the reference speed ω_k.The upper-bound intervals are [ω_k, ∞) or [ω_k, 0], while the lower-bound intervals are [0, ω_k] or (−∞, ω_k].
  • Appendix: The first branch yields explicit roots such as ω_U = ω_k + 1.5pΦ_pmI_maxΔγ/J and ω_L = ω_k − 1.5pΦ_pmI_maxΔγ/J.
  • Appendix: The third-branch real root is obtained explicitly, with corresponding upper-bound roots yielding the stated closed-form expressions.
  • Appendix: The second branch is transformed by squaring into a sixth-order polynomial, whose coefficients are shared by the lower- and upper-bound cases.The shared coefficients result from applying the square operation to the boundary equation.

B. Proof of Lemma 2

The proof constructs complete boundary-root sets by analyzing the piecewise-analytic torque function and shows that augmenting them produces a finite set containing the optimum.

  • B. Proof of Lemma 2: The motor torque function is piecewise analytic with at most three branches, so the nonlinear boundary equation is analyzed separately on each branch.
  • B. Proof of Lemma 2: For the first and third branches, explicit candidate roots are filtered using the corresponding range constraints on ω.
  • B. Proof of Lemma 2: For the second branch, all real roots of the sixth-order polynomial are obtained from real eigenvalues of its companion matrix and then range-filtered.
  • B. Proof of Lemma 2: Algorithm 1 combines valid roots into complete boundary sets, while Algorithm 2 augments them into a finite candidate set guaranteed to contain the optimal solution.
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