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From Multi-Modal Paths to Executable Trajectories: A Trajectory Planning Framework for 4WIS Robots

Runjiao Bao, Lin Zhang, Yongkang Xu, Shoukun Wang

arXiv:2608.29108v1cs.RO

TL;DR

Planning a 4WIS robot requires a feasible trajectory from an initial to a goal configuration in free space. The paper combines mode-aware global search with segment-wise trajectory optimization; experiments report safe, mode-conforming trajectories with zero terminal pose error and the shortest arrival and computation times among compared planners.

Abstract

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Four-wheel independent steering (4WIS) mobile robots support multiple motion modes, offering high maneuverability in narrow and complex environments. However, existing planning methods often fail to fully exploit these capabilities, leading to suboptimal trajectory quality. To address this limitation, this paper proposes a multi-modal global trajectory planning framework that couples mode-augmented front-end search with mode-consistent segment-wise trajectory optimization. In the front-end stage, Hybrid A* is extended to a four-dimensional state space incorporating motion modes, while mode-switching-aware cost and heuristic functions embed mode decisions into the global search process. Multi-modal Reeds-Shepp curves and an intelligent terminal connection strategy are further designed to improve search efficiency. In the back-end stage, a segment-wise trajectory optimization framework based on an improved iterative safe corridor scheme is developed to convert discrete multi-modal paths into smooth, kinematically feasible trajectories with stationary mode transitions. Experimental results show that the proposed method achieves the best overall performance in safety, arrival time, terminal accuracy and computation time. Real-world experiments on a physical 4WIS robot further validate the practical effectiveness and executability of the generated trajectories, providing a flexible and high-performance solution for multi-modal mobile robot trajectory planning.

I. INTRODUCTION … III. PRELIMINARIES

The paper frames a gap between multi-modal path planning and executable 4WIS trajectories, then proposes a framework that integrates mode decisions into global search and trajectory optimization.

I. INTRODUCTION

  • Existing planning often defers mode choice to execution, risking uncertainty and transient instability; prior multi-modal planners typically produce discrete paths rather than executable trajectories.The introduction motivates restricting mode transitions to near-zero velocity states.Source: §I · p. 1
  • The proposed framework embeds mode decisions in both global path search and continuous trajectory optimization rather than deferring them to execution.The paper reports simulation and real-world experiments to validate the framework.Source: §I · p. 1
  • Mode-augmented Hybrid A* treats motion mode as an explicit discrete planning variable and evaluates intra-mode primitives and inter-mode switching candidates in one search.Source: §I · p. 1
  • The front end incorporates mode-dependent terminal connections, switching costs, and a corresponding heuristic into node scoring to guide mode selection during global search.Source: §I · p. 1
  • The back-end optimizer inherits front-end mode labels and switching nodes, applying mode-specific dynamics, shared switching states, and near-zero switching velocities.These constraints are specified to ensure physical executability.Source: §I · pp. 1–2

A. Mobile Robots Motion Planning

  • Search-based methods can neglect kinematics, sampling offers uncertain solution quality, optimization depends on initialization, and learning faces training-cost, generalization, and sim-to-real challenges.Source: A. Mobile Robots Motion Planning · p. 2

B. 4WIS Robots Motion Planning

  • Prior 4WIS research largely addresses single-mode modeling and control,,, despite the hybrid system's continuous inputs and discrete mode selections.Source: B. 4WIS Robots Motion Planning · p. 2
  • Execution-stage methods,,, defer mode choice, while planning-stage methods, consider modes but leave a gap to executable, constraint-satisfying trajectories.Source: B. 4WIS Robots Motion Planning · p. 2

A. Trajectory Planning Problem Statement · B. 4WIS Robots Motion Modes

The planning problem is to find a feasible collision-free trajectory between specified configurations. The 4WIS model represents its three motion modes through an equivalent bicycle model and mode-dependent kinematics.

A. Trajectory Planning Problem Statement

  • The planner seeks a feasible trajectory through collision-free configurations from the specified initial configuration to the goal.The workspace contains static obstacles, and the robot configuration consists of planar position and heading.Source: A. Trajectory Planning Problem Statement · p. 2
  • A practical path uses discrete waypoints whose transitions satisfy kinematic limits and whose configurations remain collision-free.The sequence begins and ends at the initial and goal configurations, respectively.Source: A. Trajectory Planning Problem Statement · p. 2

B. 4WIS Robots Motion Modes

  • The three modes are Ackermann, lateral steering, and parallel movement; Ackermann suits open-space driving and turning, while lateral steering supports parking and lateral pose adjustment.Ackermann constrains the instantaneous center of rotation to the body’s lateral axis; lateral steering constrains it to the longitudinal axis.Source: B. 4WIS Robots Motion Modes · p. 3
  • When tire slip is negligible, the four-wheel steering structure is approximated by an equivalent bicycle model, with virtual steering and center velocity mapped to physical wheels using double-Ackermann steering.The equivalent model's steering angle and velocity represent the virtual wheel and robot center, not individual physical wheels.Source: B. 4WIS Robots Motion Modes · p. 3
  • The system state and controls describe position, heading, signed center velocity, equivalent steering angle, acceleration, and steering rate, with mode-specific body kinematics and a shared control input matrix.The discrete model uses time step ∆t; the control input matrix is identical across modes.Source: B. 4WIS Robots Motion Modes · p. 3
  • In parallel movement, all wheels share steering angles and velocities, placing the instantaneous center of rotation at infinity so the robot translates at constant heading for obstacle avoidance, docking, or alignment.Source: B. 4WIS Robots Motion Modes · p. 3

C. Collision Model

The collision model approximates the robot with two equal-radius circles along its longitudinal axis and checks their centers against obstacles inflated by the footprint radius plus a safety margin.

C. Collision Model

  • The robot footprint is represented by equal-radius front and rear circles placed along its longitudinal axis, with centers computed from state (x, y, θ).The representation is intended to simplify geometric modeling and improve computational efficiency.Source: C. Collision Model · p. 3
  • Collision avoidance requires both circle centers to lie outside obstacles dilated by a disk of radius rc = rb + rsafe.The dilation uses a Minkowski sum, where rb is the circle radius and rsafe ≥ 0 is an optional safety margin.Source: C. Collision Model · p. 3
  • The safety margin rsafe was set to 0.02 m in all experiments.The resulting compact formulation enables efficient collision checking while conservatively accounting for the robot footprint.Source: C. Collision Model · p. 3

IV. METHODOLOGY · A. Front-end Path Generation · 1) Multi-Modal State Representation:

The framework combines front-end path generation with back-end trajectory optimization for 4WIS robots. Its front end extends Hybrid A* with explicit motion modes and mode-aware expansion and switching.

IV. METHODOLOGY

  • The framework integrates front-end path generation with back-end trajectory optimization to seek collision-free, kinematically feasible navigation in complex environments.Source: §IV · p. 4

1) Multi-Modal State Representation:

  • Hybrid A* is adopted for its modular motion primitives, heuristic, and terminal-connection module; near the goal, it attempts a collision-checked Reeds–Shepp connection.A collision-free connection returns a solution; otherwise, regular search expansions continue.Source: 1) Multi-Modal State Representation · p. 4
  • The search state extends traditional Hybrid A* from (x, y, θ) to (x, y, θ, m), explicitly modeling motion mode to represent 4WIS maneuverability.Traditional Hybrid A* is described as designed for fixed motion patterns, typically Ackermann steering.Source: 1) Multi-Modal State Representation · p. 4
  • The mode variable m identifies Ackermann steering, lateral steering, or parallel movement; in parallel mode, heading stays fixed and common wheel angle ϕ sets direction.Source: 1) Multi-Modal State Representation · p. 4
  • Primitive distance is computed as Δs = vref × Δt from constant reference velocity and sampling time, replacing fixed forward iteration distance to align lengths across modes.This mechanism supports subsequent multi-modal Reeds–Shepp expansion and mode-switching cost quantification.Source: 1) Multi-Modal State Representation · p. 4
  • Intra-mode expansion preserves the current mode, using mode-specific primitives; Ackermann and lateral steering update heading by yaw-rate models, while parallel motion uses wheel angle ϕ for translation.Both forward and backward driving directions are considered.Source: 1) Multi-Modal State Representation · p. 4
  • Inter-mode expansion inserts a switching node at the current pose and changes only the mode label, representing a stop for reconfiguration rather than an instantaneous switch while moving.Switching cost accounts for stopping and reconfiguration; back-end segment-wise optimization generates continuous deceleration and acceleration.Source: 1) Multi-Modal State Representation · p. 5

2) Multi-Modal Reeds-Shepp Curves: · 3) Cost and Heuristic Functions:

The framework adapts Reeds–Shepp connections to 4WIS motion modes and incorporates mode-switching costs into Hybrid A* cost and heuristic evaluation. Its unified scoring evaluates feasible modes without rule-based pruning.

2) Multi-Modal Reeds-Shepp Curves:

  • The curves use mode-specific curvature bounds: 2 tan ϕmax/L for Ackermann, 2 tan ϕmax/W for lateral steering, and ϕ̇max/vref for parallel translation.The first two bounds are geometric; the parallel-translation bound depends on translational speed.Source: 2) Multi-Modal Reeds-Shepp Curves · p. 5
  • Lateral steering reuses standard Reeds–Shepp construction through a coordinate transform, while parallel translation is activated only below a prescribed heading-error tolerance.The lateral transform treats track width W as the equivalent wheelbase and maps the resulting curve back to the original frame.Source: 2) Multi-Modal Reeds-Shepp Curves · p. 5

3) Cost and Heuristic Functions:

  • A mode switch is costed as velocity reconfiguration plus switching latency, and an initial-node switch incurs only the acceleration phase, halving the velocity-related term.The latency is set by tswitch and the acceleration term uses amax.Source: 3) Cost and Heuristic Functions · p. 5
  • The accumulated cost adds steering-change, direction-change, and mode-switch penalties; at switching nodes, only Cswitch is retained and all other terms are zero.The mode-switch penalty is applied when the motion mode differs from the parent node's mode.Source: 3) Cost and Heuristic Functions · p. 5
  • The heuristic combines Euclidean distance with the Reeds–Shepp connection cost for the corresponding motion mode, retaining Hybrid A* admissibility.The formulation uses the available motion modes when constructing the heuristic.Source: 3) Cost and Heuristic Functions · p. 5
  • The unified score f(n) = g(n) + h(n) evaluates feasible successors across modes without separate rule-based mode pruning.Both accumulated cost and heuristic include the corresponding mode-switching penalty, so lower-cost modes are prioritized while other feasible modes remain available for later expansion.Source: 3) Cost and Heuristic Functions · p. 5

4) Terminal Connection Strategy: · B. Back-end Path Optimization · 1) Construction of Safe Corridors:

The framework selects cost-minimizing, collision-free multi-modal terminal connections, then converts the discrete path into a continuous trajectory using obstacle-aware safe corridors. Its adaptive corridor construction guides expansion by motion direction and supplies constraints for trajectory optimization.

4) Terminal Connection Strategy:

1) Construction of Safe Corridors:

  • Safe corridors provide a restricted but connected optimization space for converting the discrete multi-modal path into a continuous, executable trajectory.The corridors are sequences of convex feasible regions around the reference path.Source: 1) Construction of Safe Corridors · p. 5

4) Terminal Connection Strategy:

  • During search, the planner generates intra-modal or inter-modal successors, accounts for switching costs, uses a multi-modal heuristic, and admits collision-free nodes.The algorithm returns failure if the open list is exhausted without finding a path.Source: 1) Construction of Safe Corridors · p. 6

1) Construction of Safe Corridors:

  • The path is resampled at spatial interval D, its orientations are made continuous, and obstacle-constrained convex regions are built around the resulting N + 1 waypoints.The corridor inclusion relation is intended to guarantee collision-free feasibility; obstacles are inflated using the geometric approximation model.Source: 1) Construction of Safe Corridors · p. 6
  • For the bi-disc robot model, the method creates front- and rear-center reference paths and synchronously builds dynamic corridors, initializing a square of side length ∆d1 at each waypoint.Source: 1) Construction of Safe Corridors · p. 6
  • Unlike fixed orthogonal expansion, the adaptive strategy partitions the workspace by local velocity direction to prioritize each corridor’s expansion direction.Source: 1) Construction of Safe Corridors · p. 6
  • Corridors expand with collision checks after each ∆d1 step, refine with ∆d2 after a collision, and stop upon another collision or reaching dmax.The expansion procedure is illustrated in Fig. 4.Source: 1) Construction of Safe Corridors · p. 6, Figure 4 · p. 6
  • The corridor bounds become linear box constraints enforced at discrete times over [0, T], guaranteeing collision avoidance while easing nonlinear-constraint difficulties in cluttered environments.The optimization horizon uses N uniform time steps with ∆t = T/N.Source: 1) Construction of Safe Corridors · p. 6

2) Iterative Optimization Framework:

The framework optimizes the front-end path as fixed-mode segments separated at predetermined mode-switching points. It preserves each switching pose and enforces stationary transitions while iteratively refining segment trajectories and safe corridors.

2) Iterative Optimization Framework:

  • The path is divided at front-end mode switches into fixed-mode segments, each initialized with a duration based on its reference length and vref.Switching points define the boundary conditions between segments.Source: 2) Iterative Optimization Framework · p. 6
  • At each mode switch, adjacent segments preserve the same geometric pose and constrain velocity to zero; concatenation inserts a mode-reconfiguration interval tswitch.The switching point’s position and orientation stay fixed while interior states and controls are optimized, so transitions occur at rest.Source: 2) Iterative Optimization Framework · p. 7
  • Each segment is optimized as an independent NLP with its mode fixed; temporally resampled and kinematically interpolated reference states and controls provide a warm start.Some constraints are relaxed through soft penalties to alleviate numerical infeasibility from strict equality and geometric-consistency constraints.Source: 2) Iterative Optimization Framework · p. 7
  • The objective combines task completion time, comfort cost ∥vk ωk∥2, and constraint-violation severity, penalizing discrete dynamics and rigid-body consistency.Front and rear circle centers are independent optimization variables, with their rigid-body relationship to vehicle pose enforced by a geometric consistency penalty.Source: 2) Iterative Optimization Framework · p. 7
  • Optimization stops when the infeasibility metric falls below εtol or reaches I iterations; otherwise, it rebuilds the safe corridor and warm-starts the next NLP from the prior solution.The infeasibility metric combines dynamic and geometric penalties.Source: 2) Iterative Optimization Framework · p. 7

V. EXPERIMENTS … 2) Evaluation Metrics:

The experiments describe the simulation platform, four scenarios, and repeated fair comparisons, then define metrics for safety, trajectory quality, terminal accuracy, motion-mode consistency, and runtime.

A. Experimental Setup

  • Simulations used an Intel i7-13700H CPU and NVIDIA RTX 4060 GPU; CasADi formulated the back-end NLP, solved by IPOPT.Table I summarizes the 4WIS platform's geometric, kinematic, and planning-related parameters.Source: A. Experimental Setup · p. 8

1) Scenario Settings:

  • The four scenarios assess detouring and mode switching, constrained terminal maneuvers, lateral motion, and trajectory generation in dense clutter.Env1 is maze-like, Env2 is structured parking, Env3 limits longitudinal clearance, and Env4 contains irregular dense obstacles.Source: 1) Scenario Settings · p. 8
  • Each planner used identical robot models and settings, with 50 repetitions per scenario and statistical analysis of planning efficiency and overall performance.Source: 1) Scenario Settings · p. 8

2) Evaluation Metrics:

  • Safety (SF) requires collision-free execution; Trajectory Length (TL) sums Euclidean distances between adjacent sampled points, and Arrival Time (AT) measures duration to the terminal state.Source: 2) Evaluation Metrics · p. 8
  • Average Curvature (AC) and Maximum Curvature (MC) measure the trajectory's average and maximum absolute curvature.Source: 2) Evaluation Metrics · p. 8
  • Terminal Position Error (TPE) is the distance from the final position to the goal, while Terminal Yaw Error (TYE) is the absolute wrapped heading error.Source: 2) Evaluation Metrics · p. 8
  • Standard-mode Rate (SMR) is the proportion of resampled samples matching stationary, parallel, Ackermann, lateral Ackermann, or in-place rotation motion; lower SMR indicates more non-standard motion and greater control difficulty.The classification threshold is η = 0.15.Source: 2) Evaluation Metrics · p. 8
  • Computation Time (CT) is the wall-clock runtime of the corresponding planning pipeline.Source: 2) Evaluation Metrics · p. 8

B. Comparison Experiments … 1) Ablation Study of the Front-End Planner:

Across four benchmark scenarios, Multi-SC balances safety, terminal accuracy, mode consistency, arrival time, and computation time; ablations show complementary roles for multi-modal primitives, RS terminal connection, and switching regulation.

B. Comparison Experiments — 1) Compared Methods:

  • The comparison includes Multi-SC, which combines mode-augmented global search with mode-consistent corridor optimization; Ackermann-only Single-SC; ICR; and sampling-based MPPI.ICR searches general 4WIS motion space before smoothing, while MPPI locally tracks a front-end reference path with obstacle-aware costs.Source: 1) Compared Methods · p. 8

B. Comparison Experiments — 2) Experimental Results Analysis:

  • Multi-SC produces safe trajectories with zero terminal position and yaw errors, SMR of 1.000, and the shortest arrival and computation times in all four scenarios.The results are summarized in Table II.Source: Table II · p. 9, 2) Experimental Results Analysis · p. 9
  • ICR has the shortest geometric paths in Env2–Env4, but its trajectories are unsafe in all four environments and its SMR ranges from 0.715 to 0.764.The back-end smoothing does not explicitly consider obstacle constraints.Source: 2) Experimental Results Analysis · pp. 9–10

C. Systematic Analysis — 1) Ablation Study of the Front-End Planner:

  • Only MP has 0.202 m terminal position error and the largest equivalent trajectory length; MP+RS removes terminal error but has 18.745 m length and the highest computation time, while MP+SCH reaches 15.500 m with 0.195 m error.Equivalent length includes the cost of mode switching; results are shown in Fig. 6 and Table III.Source: Figure 6 · p. 10, Table III · p. 10, 1) Ablation Study of the Front-End… · p. 10

B. Comparison Experiments — 2) Experimental Results Analysis:

  • MPPI produces collision-free trajectories in every scenario, but has the longest arrival times, higher computation time than both SC methods, terminal errors, and SMR of 0.720–0.904.It uses local sampling and tracking without explicitly enforcing an exact terminal-state constraint; some segments contain non-standard coupled motions.Source: 2) Experimental Results Analysis · p. 10
  • Multi-SC reduces trajectory length, arrival time, and computation time versus Single-SC by averages of 4.4%, 6.7%, and 30.6%, respectively.Lateral or parallel motion avoids Ackermann-only detours; mode-dependent terminal connection and fixed-mode segment optimization improve search efficiency and reduce optimization complexity.Source: 2) Experimental Results Analysis · p. 10

C. Systematic Analysis — 1) Ablation Study of the Front-End Planner:

  • Full achieves zero terminal position and yaw errors, the shortest equivalent trajectory length, and the lowest computation time among ablations.Multi-modal primitives provide reachability, RS terminal connection improves accuracy, and switching heuristic and cost improve mode selection; together they yield a more compact, accurate, efficient front-end plan.Source: Table III · p. 10, 1) Ablation Study of the Front-End… · p. 10
  • In cluttered Env4, the ablation compares Only MP, MP+RS, MP+SCH, and Full, successively testing multi-modal primitives, RS terminal connection, and switching regulation.The variants selectively disable terminal connection or mode-switching heuristic and cost; Full enables all modules.Source: 1) Ablation Study of the Front-End… · p. 10

2) Sensitivity to the Reference Velocity: · D. Results of Real-world Experiments · VI. CONCLUSION

These sections explain how reference velocity shapes motion-mode preferences, report real-world planning and execution results, and summarize the framework’s outcomes and future extensions.

2) Sensitivity to the Reference Velocity:

  • At vref = 1.0 m/s the planner combined Ackermann, lateral, and parallel motions; at 1.5 m/s it suppressed unnecessary transitions while retaining lateral motion; at 2.0 m/s it favored Ackermann motion.The three reference velocities were evaluated in the same structured scenario; Fig. 7 shows their effect on mode switching.Source: 2) Sensitivity to the Reference Velocity · p. 11, Figure 7 · p. 11
  • Smaller vref values encourage more frequent use of different motion modes, whereas larger values favor fewer mode transitions and stronger mode consistency.Reference velocity affects the relative mode-switching cost in front-end search, making it an interpretable trade-off between multi-modal flexibility and transition conservativeness.Source: 2) Sensitivity to the Reference Velocity · pp. 10–11

D. Results of Real-world Experiments

  • Real-world tests placed static obstacles in the environment, used an LIO-SAM global map, and required the platform to plan and track a collision-free trajectory to the target.Platform parameters followed Table I.Source: D. Results of Real-world Experiments · p. 11
  • 15.09 s for the multi-modal trajectory versus 18.46 s for the single-modal trajectory; front-end and back-end planning took 0.82 s and 0.57 s, totaling 1.39 s.The complete global trajectory was generated before execution.Source: D. Results of Real-world Experiments · p. 11, Figure 8 · p. 11
  • 0.0096 m average and 0.0388 m maximum lateral errors, and 0.0086 m average and 0.0293 m maximum longitudinal errors, were measured after excluding mode-switching intervals.A kinematic-model MPC controller tracked the trajectory in real time at approximately 20 Hz.Source: D. Results of Real-world Experiments · p. 11, Figure 8 · p. 11

VI. CONCLUSION

  • The framework extends Hybrid A* with explicit motion-mode modeling, mode-switch-aware costs and heuristics, multi-modal Reeds-Shepp curves, and an intelligent terminal connection strategy.Its back end uses mode-consistent segment-wise trajectory optimization with an improved iterative safe corridor scheme to produce smooth, kinematically feasible trajectories with stationary mode transitions.Source: §VI · pp. 11–12
  • The framework generated safe trajectories conforming to predefined motion modes, achieved zero terminal pose error, and had the shortest arrival and computation times among compared planners.Real-world experiments further verified the generated trajectories’ executability and practical effectiveness.Source: §VI · p. 12
  • Future work will optimize the necessity and locations of mode-switching points and extend safe corridors from axis-aligned boxes to general convex polytopes.The proposed extensions aim to improve trajectory quality and reduce corridor conservativeness while preserving feasibility.Source: §VI · p. 12

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