Timed elastic band

Overview

A local planner decides how the robot should move now, around nearby obstacles. The timed elastic band (TEB) represents the trajectory as a band of poses between the robot and its goal, with a time interval between each pair of poses. The band is adjusted by minimising a cost. The cost adds the squared time intervals to weighted penalties that are active only when a kinematic, clearance, velocity or acceleration limit is violated 2411.01814 §2.2 2411.01814 eq. 2. Both the start and the goal stay fixed during optimisation 2411.01814 eq. 3.

The two cards in this set both extend TEB and keep its optimiser. They differ in what they add. MP-TEB adds predicted human motion to the cost 2411.01814 §2. RTEB adds a recovery path for the cases where the TEB trajectory cannot be driven 2412.03174 §3.

Variants

Base TEB cost

The base form is the cost in 2411.01814 eq. 2. The paper optimises several locally optimal bands for different obstacle configurations and keeps the cheapest one 2411.01814 eq. 3 2411.01814 eq. 4. The weight values and the maximum time interval are not stated in that paper 2411.01814 §2.2.

Social extension: MP-TEB

The forecaster is a Social GAN. It takes the last 8 positions of each tracked person, sampled every 0.5 s, and predicts 12 future positions 2411.01814 §2.1. The planner adds three social terms to the base cost: a term that pulls toward a predicted human path, a term for a predicted dynamic obstacle that moves at its velocity, and a priority-side term that is subtracted 2411.01814 eq. 5 2411.01814 eq. 6 2411.01814 eq. 7. The total is the base cost plus a scaled social cost 2411.01814 eq. 8. The authors report it in Gazebo against DWA and TEB on four scenarios, with the shortest total time in three of them 2411.01814 tab. 1. Card: Nguyen Canh et al. (MP-TEB).

Recovery layer: RTEB

RTEB checks the kinematic feasibility of the current trajectory at each iteration. The paper does not give the test or its threshold 2412.03174 §3. When the trajectory is infeasible, a hybrid A* search over car-like states builds a drivable path 2412.03174 §3.1.1. A Voronoi-field obstacle term and curvature and path terms then smooth that path, with weights 0.5, 0.3 and 0.2 2412.03174 §3.1.2 2412.03174 eq. 1. TEB restarts from the smoothed path 2412.03174 §3. Near the goal, a pre-goal pose aligns the heading 2412.03174 §3.1.3. On dense obstacle scenes the authors report lower traverse distance, time and control effort than TEB and NMPC at 30% to 50% density 2412.03174 tab. 4. Card: Kulathunga et al. (RTEB).

When to use which

Implementation sketch

The common core is a band optimisation with fixed endpoints. Each variant adds its own step around it. Steps marked as optional are specific to one variant.

# TEB family control step  [arxiv:2411.01814 §2.2] [arxiv:2412.03174 §3]
# Input:  band of poses, goal pose, optional person tracks
# Output: optimised band; a command must still be derived from it

def teb_step(band, goal, tracks):
    # Optional recovery (RTEB): replace infeasible band  [arxiv:2412.03174 §3]
    if not kinematically_feasible(band):
        path = hybrid_astar(band[0], goal)              # [arxiv:2412.03174 §3.1.1]
        band = reinitialise(smooth(path))               # [arxiv:2412.03174 §3.1.2]
    # Base cost: time intervals plus limit penalties  [arxiv:2411.01814 eq. 2]
    cost = teb_cost(band)
    # Optional social cost (MP-TEB): added with a scale factor  [arxiv:2411.01814 eq. 8]
    if tracks is not None:
        cost += delta_mp * social_cost(band, tracks)    # sign: see Contradictions
    # Minimise with the endpoints fixed  [arxiv:2411.01814 eq. 3]
    return minimise(cost, band, fix_endpoints=True)

Contradictions

• Sign of the social terms. As printed, the human-path and dynamic-obstacle terms grow with distance and are added to the cost. Read literally, the planner is pulled toward those points, which is the opposite of the avoidance the paper describes 2411.01814 eq. 5 2411.01814 eq. 6 2411.01814 §2.2. The paper does not settle the sign.
• Which cost is optimised. The local optimum is written with the base cost, while the social cost is defined separately 2411.01814 eq. 3 2411.01814 eq. 8.
• Text against table, MP-TEB. The text says MP-TEB is better on efficiency and safety in every scenario. In the corridor scenario DWA has the shorter total time and TEB the shorter path 2411.01814 tab. 1 2411.01814 §3.2.
• Text against table, RTEB. The text says RTEB has lower values across all metrics. In goal-alignment scenario 3, RTEB's distance is larger than TEB's (5.95 m against 5.41 m) 2412.03174 tab. 2 2412.03174 §4.3.
• Headline figure. The abstract gives about 20% lower distance, time and control effort. No single table value matches it. Derived by the compiler from the table, not stated by the authors: at 30% to 50% density the distance gain is 10% to 33% 2412.03174 tab. 4.
• Planning frequency. RTEB's TEB frequency is 13 Hz in one table and 15 to 20 Hz in another, and the two tables use different definitions 2412.03174 tab. 3 2412.03174 tab. 4 2412.03174 §4.3.
• Different benchmarks. MP-TEB is compared with DWA and TEB in Gazebo on four scenes 2411.01814 tab. 1. RTEB is compared with TEB and NMPC on dense farm-like scenes in simulation and on a real vehicle 2412.03174 tab. 4. The gains are not measured on a common benchmark.

Papers

• Nguyen Canh et al. (MP-TEB): adds Social GAN predictions and three social cost terms to TEB.
• Kulathunga et al. (RTEB): adds a hybrid A* and Voronoi-smoothing recovery layer that restarts TEB.