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A bioinspired internal model-based online estimator for planar pursuit

Tengyue Liu, Xincheng Li, Sofia Morales Ferreira, Kevin Galloway, Udit Halder

arXiv:2609.25470v1cs.ROeess.SY

TL;DR

Pursuit feedback relies on relative quantities that onboard sensors may not provide, including target heading and motion commands. The paper develops an internal model-based estimator that reconstructs these quantities through optimal control and a moving-window solution. Simulations show improved estimation over conventional estimators, and robotic experiments demonstrate reconstruction and use of estimates in two pursuit strategies.

  • Problem

    Pursuit feedback may require target heading and motion controls that are generally unknown to the pursuer and must be estimated from observed relative motion.

    Source: C. Practical limitations of… · p. 2

  • Method

    An internal model constrained by relative kinematics reconstructs relative configuration and unknown target motion from partial observations using an optimal-control formulation and moving-window implementation.

    Source: §I · p. 1, §V · p. 7

  • Results

    Numerical simulations show improved estimation relative to conventional estimators, while robotic experiments demonstrate reconstruction of unmeasured relative heading and target motion from LiDAR measurements.

    Source: §V · p. 7

  • Takeaways & Limitations

    The resulting estimates are incorporated into feedback laws for constant-bearing pursuit and mutual motion camouflage.

    Source: §V · p. 7

    Pursuit feedback quantities may not be directly available to an individual agent; from the pursuer's perspective, target heading and speed and steering controls are generally unknown.

    Source: C. Practical limitations of… · p. 2

Abstract

from arXiv · show

Bioinspired feedback controls for pursuit, tracking, and collective motion are often expressed in terms of the relative configuration between interacting agents. In practice, however, onboard sensors may not directly provide all quantities required for feedback control, necessitating estimation of unobserved quantities. This paper develops a bioinspired internal model-based estimator for reconstructing those quantities from partial sensory observations and known self-motion. State reconstruction is posed as an optimization problem that treats the relative kinematics as constraints and minimizes the disagreement between the internal model outputs and measurements from onboard sensors. Pontryagin's Maximum Principle is used to derive the necessary optimality conditions. A forward-backward algorithm is used to provide a numerical solution and a moving horizon formulation is employed for online implementation. The estimator is evaluated numerically against classical state estimators. Real-time implementation of the proposed framework on robotic hardware is demonstrated through two pursuit strategies.

I. INTRODUCTION

Engineered pursuit strategies use relative geometry, but pursuers may lack measurements of quantities their controllers require. This paper introduces an internal model-based estimator to reconstruct relative configuration and unknown target motion from partial observations, then demonstrates its use in pursuit control.

I. INTRODUCTION

  • Natural pursuit includes constant-bearing and motion-camouflage strategies, which have also inspired engineered pursuit and interception systems.Engineered feedback commonly uses relative position, bearing, and prey turning rate, which may not be available directly in practice.Source: §I · p. 1
  • The estimator optimizes an internal target trajectory constrained by known relative kinematics to explain onboard measurements.The framework is presented in the planar setting for simplicity.Source: §I · p. 1
  • The internal model-based estimator reconstructs full relative configuration and unknown target motion commands from partial, intermittent observations such as distance and bearing.Its optimal-control formulation uses Pontryagin’s Maximum Principle; the paper highlights estimating linear and angular speeds of a freely moving target.Source: §I · p. 1
  • An iterative algorithm solves the optimality conditions, and a moving-horizon implementation supports online estimation.Source: §I · p. 1
  • Numerical simulations and robotic experiments demonstrate using the estimates in constant-bearing pursuit and mutual motion camouflage.Source: §I · p. 1

A. Agent model

  • Each agent is modeled as a unicycle with translational speed and angular speed as its control inputs.Agent 2 is the pursuer and agent 1 the pursuee unless otherwise specified.Source: A. Agent model · p. 2

B. Relative motion (plant model)

The plant model describes the pursuee's position and heading relative to the pursuer, while pursuit feedback laws may require quantities the pursuer cannot directly observe. Consequently, unknown target heading and motion controls must be estimated from observed relative motion for those laws to be executed.

B. Relative motion (plant model)

  • The relative state is the pursuee's position and heading in the pursuer's body-fixed frame, represented by z = [x, y, θ]ᵀ.The coordinates also admit an SE(2) relative rigid-body transformation interpretation.Source: B. Relative motion (plant model) · p. 2
  • The pursuer observes relative position (x, y), while the relative dynamics serve as the plant model.Source: B. Relative motion (plant model) · p. 2

C. Practical limitations of implementing feedback control

  • Pursuit feedback may require the pursuee's heading, speed, and steering controls, which are generally unknown to the pursuer and therefore must be estimated from observed relative motion.Source: C. Practical limitations of… · p. 2

III. INTERNAL MODEL-BASED ESTIMATOR (IME)

The IME uses partial observations to reconstruct relative states and unknown target controls needed for feedback. It represents the target through an internal model and formulates estimation as a constrained dynamic optimization problem.

A. Internal model and estimation

  • The pursuer’s internal model estimates relative state and unknown target controls, distinguishing the IME from observers that assume known inputs or require a target-control evolution model.The pursuer’s own control enters the internal dynamics directly.Source: A. Internal model and estimation · pp. 2–3

B. Estimation as an optimization problem

  • The optimization constrains the augmented internal state by its dynamics and uses target-control derivatives as decision variables so reconstructed controls remain continuous between observations.The decision variables are the rates of change of the estimated target’s linear and angular controls.Source: B. Estimation as an optimization problem · p. 3
  • The objective penalizes disagreement between internal-model outputs and measurements, while regularizing changes in estimated target inputs; Pontryagin’s Maximum Principle supplies necessary optimality conditions,.Source: B. Estimation as an optimization problem · p. 3

C. Necessary optimality conditions

The paper applies Pontryagin’s Maximum Principle to the estimation problem. Its necessary conditions specify state and costate evolution, observation-time jump conditions, and pointwise control selection.

C. Necessary optimality conditions

  • The optimal augmented state and its costate satisfy coupled differential equations, with jump and boundary conditions imposed at observation times.Source: C. Necessary optimality conditions · p. 3
  • The optimal control is selected by pointwise maximization of the pre-Hamiltonian, while the costate jump condition is tied to the observation mismatch at each measurement time.Source: C. Necessary optimality conditions · p. 3
  • The necessary conditions for the minimization problem follow from a standard application of Pontryagin’s Maximum Principle with discrete-time observation costs [27, Theorem 2.2].Source: C. Necessary optimality conditions · p. 3

D. Numerical algorithms

This section describes numerical algorithms for solving the estimation optimization problem and implementing it online.

D. Numerical algorithms

1) Forward-backward Algorithm:

The estimator’s forward-backward algorithm iteratively solves the optimization problem, while a moving horizon formulation reuses recent observations and the preceding solution for online operation.

1) Forward-backward Algorithm:

2) Online implementation using moving horizon:

  • Repeatedly solving the optimization from an arbitrary initial guess is impractical for real-time feedback because iterative forward-backward sweeps may be numerous.Source: 2) Online implementation using moving… · p. 4
  • The moving horizon uses the latest Nh observations, shifts the window as new data arrive, and initializes each solve from the preceding window’s solution.The approach is related to model predictive control, but uses past measurements over a receding window.Source: 2) Online implementation using moving… · p. 4

IV. NUMERICAL AND EXPERIMENTAL RESULTS

The evaluation considers estimator accuracy and control execution, focusing on constant-bearing pursuit and mutual motion camouflage. Both strategies are expressed through relative configuration variables and specific desired manifolds.

IV. NUMERICAL AND EXPERIMENTAL RESULTS

  • The evaluation measures estimator accuracy and control execution; the code for simulations and experiments is available at.Source: §IV · p. 4

A. Bioinspired feedback controls for pursuit

1) Constant Bearing (CB):

  • Constant-bearing pursuit steers agent 2 toward a prescribed bearing ϕ, making the manifold α2 = ϕ invariant and attractive.Source: 1) Constant Bearing (CB) · p. 4

2) Mutual Motion Camouflage (MMC):

  • In mutual motion camouflage, both agents pursue one another using constant speeds and steering controls that make the energy-level manifold E = E0 invariant and locally attractive.Source: 2) Mutual Motion Camouflage (MMC) · p. 4

B. Numerical simulation results

The numerical comparison evaluates open-loop estimates of relative configuration and target controls against EKF and PF baselines. After the initial transient, IME has lower reported errors on key estimates and less sensitivity to initialization in relative-pose estimation.

B. Numerical simulation results

  • IME uses MATLAB ode45 with a forward-backward solver and moving-horizon initialization; IME and EKF were tested with 100 random initializations, while PF used 35,000 particles per trial.Source: B. Numerical simulation results · pp. 4–5
  • The comparison uses EKF and PF baselines that assume zero dynamics for target controls; all estimators receive the same synthetic LiDAR measurement-noise sequence.Source: B. Numerical simulation results · pp. 4–5
  • Open-loop tests estimate relative configuration and target controls along a prescribed trajectory, using relative-pose, translational-velocity, and angular-velocity errors.Source: 1) Experiment A · p. 5
  • After the initial transient, IME has the lowest median eg and ev1, narrower eg percentile bounds, and lower converged eu1 than EKF and PF.Source: Figure 3 · p. 5, 1) Experiment A · p. 5

2) Experiment B:

In simulation, the IME supports constant-bearing pursuit with lower median error and narrower variability than the EKF and PF; robot experiments show rapid establishment of the desired bearing and reconstruction of target motion.

2) Experiment B:

  • In Experiment B, the pursuee follows time-varying speed and steering commands, the pursuer moves at constant speed, and runs stop at 0.25 m separation.The constant-bearing controller uses μ = 2 and desired bearing 20°.Source: 2) Experiment B · p. 5
  • The IME achieves lower median constant-bearing error, a smaller percentile range, and better reconstruction of time-varying inputs than the EKF and PF in closed-loop simulation.All three estimators reduce the constant-bearing error and keep it near zero after the initial reduction.Source: 2) Experiment B · p. 5, Figure 4 · p. 6

C. Robotic experimental results

  • Two TurtleBot3 Burger robots use front and rear LiDARs for onboard sensing, while OptiTrack records ground truth only for post-experiment evaluation.The LiDAR scans are merged before target extraction; each provides an approximately 270° effective field of view.Source: C. Robotic experimental results · pp. 5–6

1) Experiment C:

  • On the TurtleBot3 robots, constant-bearing error falls near zero within the first few seconds and remains close to zero as the pursuer intercepts the pursuee.The experiment uses IME estimates to calculate the pursuer's steering control.Source: 1) Experiment C · p. 6
  • Translational relative-pose error remains small, while angular error is larger throughout the robot experiment.The translational and angular errors are shown on the left and right vertical axes, respectively.Source: 1) Experiment C · p. 6
  • The IME estimate of pursuee linear speed closely follows its command, while the angular-speed estimate deviates more but captures the command's time-varying trend.Source: 1) Experiment C · p. 6

2) Experiment D:

In mutual motion camouflage, both agents estimate local relative states and steer independently; the observed trajectories approach the desired orbit, while relative-pose errors fluctuate at roughly consistent magnitudes. The conclusion identifies observability, efficiency, and broader dynamics as future work.

2) Experiment D:

  • Both agents estimate relative states locally from their own LiDAR measurements and independently generate steering commands for mutual motion camouflage.Agent 1 reconstructs α2 and η2 from (ρ, α1); agent 2 reconstructs α1 and η1 from (ρ, α2).Source: 2) Experiment D · p. 6
  • The trajectories exhibit coordinated motion, and the phase portrait approaches the desired orbit corresponding to E = E0.The phase portrait initially follows an outward spiral.Source: 2) Experiment D · p. 7
  • The mutual-motion-camouflage metric falls from approximately 0.8 and fluctuates around 0.1 after the initial transient; its peaks vary with inter-agent separation.Larger peaks occur near maximum separation and smaller peaks near minimum separation, suggesting sensing and reconstruction errors vary with relative geometry.Source: 2) Experiment D · p. 7
  • Each agent's relative-pose error fluctuates, but the overall magnitudes remain approximately consistent throughout the experiment.The errors are computed against OptiTrack-derived ground truth in each agent's body-fixed frame.Source: 2) Experiment D · p. 7

V. CONCLUSION AND FUTURE WORK

  • The conclusion describes an internal-model estimator solved by a forward-backward algorithm with a moving window, and reports numerical and robotic demonstrations of estimation and feedback use.The estimates are incorporated into constant-bearing pursuit and mutual motion camouflage feedback laws.Source: §V · p. 7
  • Unique-recovery conditions, computational efficiency, convergence, parameter sensitivity, noisy measurements, constrained inputs, broader target dynamics, and multi-agent or three-dimensional extensions remain future work.Source: §V · p. 7

APPENDIX I DETAILS OF THE PMP NECESSARY CONDITIONS

The appendix expands the costate dynamics and describes how observation discrepancies enter the optimality conditions through costate jumps.

APPENDIX I DETAILS OF THE PMP NECESSARY CONDITIONS

  • The costate dynamics evolve the position and heading costates using the pursuer's angular speed and estimated target heading and speed.The expanded equations also specify the dynamics for the speed and angular-speed costates.Source: Appendix I · p. 8
  • The forward-backward update rule specifies how the decision variables are updated.Source: Appendix I · p. 8
  • Measurement-versus-estimate error enters the framework through the costate jump conditions at observation times.Source: Appendix I · p. 8

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