fix(vol4): use standard bracketed citation syntax in chapter 11 figures

Author: profvjreddi

books/vol4/chapters/11-planning/11-planning.qmd

@@ -136,7 +136,7 @@ Size the action chunk horizon $N$ against the mission tail quantile $Q_{1-\epsil
 
 When a generative policy serves as the planner, iterative score evaluation and denoising steps govern the shape of this latency distribution. The internal mechanics of diffusion schedules, attention caching, and token generation sit below the first budgetable quantity of the physical AI system. The systems interface treats the learned model strictly as a stochastic producer of motion chunks characterized by an empirical latency profile and an output horizon $N$. In physical hardware deployments of receding-horizon action chunking, the robot predicts a trajectory chunk of horizon $T_p$, executes a committed subset of steps $T_a$, and replans on subsequent sensor ingress, allowing the controller to absorb inference latency while reacting dynamically to disturbances (@fig-11-action-chunking-trace). Sizing $N$ against the latency tail guarantees that a replacement chunk is resident in memory before the active plan runs out of samples. Yet ensuring that a replacement trajectory arrives in time addresses only the temporal continuity of the plan. The state samples contained within the admitted chunk must also match the kinematic capabilities of the joints and remain continuous with the physical velocity and acceleration of the mechanism at the handoff seam.
 
-![Receding-Horizon Action Chunking and Closed-Loop Disturbance Recovery on Real Robot Hardware. Hardware deployment of a receding-horizon diffusion policy ($T_p = 16$ predicted action steps, $T_a = 8$ executed steps before replanning) on a canonical 6-DOF industrial manipulator performing the planar Push-T manipulation task under dynamic perturbations. (Left) Visual Occlusion: A human hand waving in front of the overhead tracking camera introduces perceptual noise and inference latency jitter, but the receding-horizon execution buffer prevents pipeline starvation and preserves continuous end-effector motion. (Middle) In-Flight State Divergence: A physical disturbance abruptly shifts the target T-block during the active pushing phase; the rolling replanning cycle detects state divergence from the recorded plan assumptions and immediately synthesizes a corrective trajectory chunk that re-acquires and aligns the block. (Right) Terminal Suffix Replanning: When the block is displaced after the manipulator has already initiated its terminal motion toward the designated end-zone, the policy aborts the terminal sequence, replans a return trajectory, and restores the block to the target alignment. Source: @chi2023diffusionpolicy.](figures/fig11_real_action_chunking_trace.png){#fig-11-action-chunking-trace fig-pos="t" width=100%}
+![Receding-Horizon Action Chunking and Closed-Loop Disturbance Recovery on Real Robot Hardware. Hardware deployment of a receding-horizon diffusion policy ($T_p = 16$ predicted action steps, $T_a = 8$ executed steps before replanning) on a canonical 6-DOF industrial manipulator performing the planar Push-T manipulation task under dynamic perturbations. (Left) Visual Occlusion: A human hand waving in front of the overhead tracking camera introduces perceptual noise and inference latency jitter, but the receding-horizon execution buffer prevents pipeline starvation and preserves continuous end-effector motion. (Middle) In-Flight State Divergence: A physical disturbance abruptly shifts the target T-block during the active pushing phase; the rolling replanning cycle detects state divergence from the recorded plan assumptions and immediately synthesizes a corrective trajectory chunk that re-acquires and aligns the block. (Right) Terminal Suffix Replanning: When the block is displaced after the manipulator has already initiated its terminal motion toward the designated end-zone, the policy aborts the terminal sequence, replans a return trajectory, and restores the block to the target alignment. Adapted from [@chi2023diffusionpolicy].](figures/fig11_real_action_chunking_trace.png){#fig-11-action-chunking-trace fig-pos="t" width=100%}
 <!-- END -->
 
 ## Feasibility and Continuity {#sec-planning-11-4}
@@ -159,7 +159,7 @@ When a generative policy serves as the planner, iterative score evaluation and d
 <!-- BEGIN -->
 A candidate trajectory produced by a learned policy or path planner is merely a sequence of geometric propositions until the nervous system verifies that the physical body can execute it. An admitted motion plan must satisfy simultaneous constraints across multiple physical domains, including joint position limits, angular velocity limits, actuator acceleration envelopes, contact force thresholds, workspace obstacle clearance, and the dynamic stopping distance required to prevent collision if an emergency occurs. A systems engineer does not compute full-body kinematics from first principles within the tracking loop [@craig2005introduction; @lynch2017modern]. Instead, the safety enforcer evaluates a machinery-specific rigid-body model that maps commanded joint positions $\mathbf{q}(t)$, velocities $\dot{\mathbf{q}}(t)$, and accelerations $\ddot{\mathbf{q}}(t)$ directly to actuator torques $\boldsymbol{\tau}(t) = \mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q})$ and motor phase currents $I_i(t) = \tau_i(t) / k_{\tau,i}$. Within-plan **feasibility** is therefore established not by geometric reachability, but by verifying that the predicted peak torque remains within the continuous thermal limit of the motor windings and the mechanical yield limits of the reduction gears across the entire time horizon. In real hardware deployments, trajectory optimization frameworks solve these constrained optimization problems over parallelized GPU threads, generating time-parameterized minimum-jerk splines that dynamically steer industrial manipulators around obstacles while strictly enforcing kinematic and torque bounds (@fig-11-manipulator-trajectory).
 
-![**Real-Time Trajectory Optimization and Dynamic Obstacle Avoidance on Industrial Manipulators.** Experimental deployment of GPU-accelerated motion generation (*cuRobo*) executing time-parameterized, minimum-jerk trajectory splines on physical hardware. (Top row) A canonical 6-DOF industrial manipulator executes an online obstacle avoidance trajectory around a dynamic workspace obstacle, continuously evaluating continuous-time collision distances against a 3D Euclidean Signed Distance Field (ESDF) voxel map generated via real-time depth perception. (Middle row) A canonical 6-DOF collaborative arm executes a collision-free pick-and-lift trajectory spline constrained by joint velocity, acceleration, and torque envelopes. (Bottom row) Coordinated multi-arm trajectory synthesis in simulation showing two canonical industrial manipulators executing synchronized, collision-free obstacle avoidance around shared workspace boundaries. Source: @sundaralingam2023curobo.](figures/fig11_real_manipulator_trajectory.png){#fig-11-manipulator-trajectory fig-pos="t" width=95%}
+![**Real-Time Trajectory Optimization and Dynamic Obstacle Avoidance on Industrial Manipulators.** Experimental deployment of GPU-accelerated motion generation (*cuRobo*) executing time-parameterized, minimum-jerk trajectory splines on physical hardware. (Top row) A canonical 6-DOF industrial manipulator executes an online obstacle avoidance trajectory around a dynamic workspace obstacle, continuously evaluating continuous-time collision distances against a 3D Euclidean Signed Distance Field (ESDF) voxel map generated via real-time depth perception. (Middle row) A canonical 6-DOF collaborative arm executes a collision-free pick-and-lift trajectory spline constrained by joint velocity, acceleration, and torque envelopes. (Bottom row) Coordinated multi-arm trajectory synthesis in simulation showing two canonical industrial manipulators executing synchronized, collision-free obstacle avoidance around shared workspace boundaries. Adapted from [@sundaralingam2023curobo].](figures/fig11_real_manipulator_trajectory.png){#fig-11-manipulator-trajectory fig-pos="t" width=95%}
 
 Validating feasibility solely at discrete planning waypoints creates a false sense of compliance. When a planner outputs trajectory points separated by a sampling interval $T_s$, the physical mechanism moves along an interpolated path between those samples. If the nervous system checks constraints only at the discrete sample timestamps $t_k = k T_s$, high-order trajectory derivatives can exceed physical bounds during the inter-sample interval $[t_k, t_{k+1}]$. Consider an analytical example of a linear joint with a maximum acceleration limit $a_{\max} = 10.0\text{ m/s}^2$ and a velocity ceiling $v_{\max} = 2.00\text{ m/s}$. If the planner outputs consecutive velocity samples $\dot{q}_k = 1.95\text{ m/s}$ and $\dot{q}_{k+1} = 1.95\text{ m/s}$ separated by $T_s = 20\text{ ms}$, a cubic spline interpolator that applies an intermediate acceleration of $a_{\max}$ produces an intra-sample velocity overshoot bounded by $\Delta v \le \frac{1}{2} a_{\max} T_s$. Evaluating this bound yields:
 $$\Delta v \le \frac{1}{2} (10.0\text{ m/s}^2)(0.020\text{ s}) = 0.10\text{ m/s}$$

publishing/quarto/contents/vol4/chapters/11-planning/11-planning.qmd

@@ -136,7 +136,7 @@ Size the action chunk horizon $N$ against the mission tail quantile $Q_{1-\epsil
 
 When a generative policy serves as the planner, iterative score evaluation and denoising steps govern the shape of this latency distribution. The internal mechanics of diffusion schedules, attention caching, and token generation sit below the first budgetable quantity of the physical AI system. The systems interface treats the learned model strictly as a stochastic producer of motion chunks characterized by an empirical latency profile and an output horizon $N$. In physical hardware deployments of receding-horizon action chunking, the robot predicts a trajectory chunk of horizon $T_p$, executes a committed subset of steps $T_a$, and replans on subsequent sensor ingress, allowing the controller to absorb inference latency while reacting dynamically to disturbances (@fig-11-action-chunking-trace). Sizing $N$ against the latency tail guarantees that a replacement chunk is resident in memory before the active plan runs out of samples. Yet ensuring that a replacement trajectory arrives in time addresses only the temporal continuity of the plan. The state samples contained within the admitted chunk must also match the kinematic capabilities of the joints and remain continuous with the physical velocity and acceleration of the mechanism at the handoff seam.
 
-![Receding-Horizon Action Chunking and Closed-Loop Disturbance Recovery on Real Robot Hardware. Hardware deployment of a receding-horizon diffusion policy ($T_p = 16$ predicted action steps, $T_a = 8$ executed steps before replanning) on a canonical 6-DOF industrial manipulator performing the planar Push-T manipulation task under dynamic perturbations. (Left) Visual Occlusion: A human hand waving in front of the overhead tracking camera introduces perceptual noise and inference latency jitter, but the receding-horizon execution buffer prevents pipeline starvation and preserves continuous end-effector motion. (Middle) In-Flight State Divergence: A physical disturbance abruptly shifts the target T-block during the active pushing phase; the rolling replanning cycle detects state divergence from the recorded plan assumptions and immediately synthesizes a corrective trajectory chunk that re-acquires and aligns the block. (Right) Terminal Suffix Replanning: When the block is displaced after the manipulator has already initiated its terminal motion toward the designated end-zone, the policy aborts the terminal sequence, replans a return trajectory, and restores the block to the target alignment. Source: @chi2023diffusionpolicy.](figures/fig11_real_action_chunking_trace.png){#fig-11-action-chunking-trace fig-pos="t" width=100%}
+![Receding-Horizon Action Chunking and Closed-Loop Disturbance Recovery on Real Robot Hardware. Hardware deployment of a receding-horizon diffusion policy ($T_p = 16$ predicted action steps, $T_a = 8$ executed steps before replanning) on a canonical 6-DOF industrial manipulator performing the planar Push-T manipulation task under dynamic perturbations. (Left) Visual Occlusion: A human hand waving in front of the overhead tracking camera introduces perceptual noise and inference latency jitter, but the receding-horizon execution buffer prevents pipeline starvation and preserves continuous end-effector motion. (Middle) In-Flight State Divergence: A physical disturbance abruptly shifts the target T-block during the active pushing phase; the rolling replanning cycle detects state divergence from the recorded plan assumptions and immediately synthesizes a corrective trajectory chunk that re-acquires and aligns the block. (Right) Terminal Suffix Replanning: When the block is displaced after the manipulator has already initiated its terminal motion toward the designated end-zone, the policy aborts the terminal sequence, replans a return trajectory, and restores the block to the target alignment. Adapted from [@chi2023diffusionpolicy].](figures/fig11_real_action_chunking_trace.png){#fig-11-action-chunking-trace fig-pos="t" width=100%}
 <!-- END -->
 
 ## Feasibility and Continuity {#sec-planning-11-4}
@@ -159,7 +159,7 @@ When a generative policy serves as the planner, iterative score evaluation and d
 <!-- BEGIN -->
 A candidate trajectory produced by a learned policy or path planner is merely a sequence of geometric propositions until the nervous system verifies that the physical body can execute it. An admitted motion plan must satisfy simultaneous constraints across multiple physical domains, including joint position limits, angular velocity limits, actuator acceleration envelopes, contact force thresholds, workspace obstacle clearance, and the dynamic stopping distance required to prevent collision if an emergency occurs. A systems engineer does not compute full-body kinematics from first principles within the tracking loop [@craig2005introduction; @lynch2017modern]. Instead, the safety enforcer evaluates a machinery-specific rigid-body model that maps commanded joint positions $\mathbf{q}(t)$, velocities $\dot{\mathbf{q}}(t)$, and accelerations $\ddot{\mathbf{q}}(t)$ directly to actuator torques $\boldsymbol{\tau}(t) = \mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{g}(\mathbf{q})$ and motor phase currents $I_i(t) = \tau_i(t) / k_{\tau,i}$. Within-plan **feasibility** is therefore established not by geometric reachability, but by verifying that the predicted peak torque remains within the continuous thermal limit of the motor windings and the mechanical yield limits of the reduction gears across the entire time horizon. In real hardware deployments, trajectory optimization frameworks solve these constrained optimization problems over parallelized GPU threads, generating time-parameterized minimum-jerk splines that dynamically steer industrial manipulators around obstacles while strictly enforcing kinematic and torque bounds (@fig-11-manipulator-trajectory).
 
-![**Real-Time Trajectory Optimization and Dynamic Obstacle Avoidance on Industrial Manipulators.** Experimental deployment of GPU-accelerated motion generation (*cuRobo*) executing time-parameterized, minimum-jerk trajectory splines on physical hardware. (Top row) A canonical 6-DOF industrial manipulator executes an online obstacle avoidance trajectory around a dynamic workspace obstacle, continuously evaluating continuous-time collision distances against a 3D Euclidean Signed Distance Field (ESDF) voxel map generated via real-time depth perception. (Middle row) A canonical 6-DOF collaborative arm executes a collision-free pick-and-lift trajectory spline constrained by joint velocity, acceleration, and torque envelopes. (Bottom row) Coordinated multi-arm trajectory synthesis in simulation showing two canonical industrial manipulators executing synchronized, collision-free obstacle avoidance around shared workspace boundaries. Source: @sundaralingam2023curobo.](figures/fig11_real_manipulator_trajectory.png){#fig-11-manipulator-trajectory fig-pos="t" width=95%}
+![**Real-Time Trajectory Optimization and Dynamic Obstacle Avoidance on Industrial Manipulators.** Experimental deployment of GPU-accelerated motion generation (*cuRobo*) executing time-parameterized, minimum-jerk trajectory splines on physical hardware. (Top row) A canonical 6-DOF industrial manipulator executes an online obstacle avoidance trajectory around a dynamic workspace obstacle, continuously evaluating continuous-time collision distances against a 3D Euclidean Signed Distance Field (ESDF) voxel map generated via real-time depth perception. (Middle row) A canonical 6-DOF collaborative arm executes a collision-free pick-and-lift trajectory spline constrained by joint velocity, acceleration, and torque envelopes. (Bottom row) Coordinated multi-arm trajectory synthesis in simulation showing two canonical industrial manipulators executing synchronized, collision-free obstacle avoidance around shared workspace boundaries. Adapted from [@sundaralingam2023curobo].](figures/fig11_real_manipulator_trajectory.png){#fig-11-manipulator-trajectory fig-pos="t" width=95%}
 
 Validating feasibility solely at discrete planning waypoints creates a false sense of compliance. When a planner outputs trajectory points separated by a sampling interval $T_s$, the physical mechanism moves along an interpolated path between those samples. If the nervous system checks constraints only at the discrete sample timestamps $t_k = k T_s$, high-order trajectory derivatives can exceed physical bounds during the inter-sample interval $[t_k, t_{k+1}]$. Consider an analytical example of a linear joint with a maximum acceleration limit $a_{\max} = 10.0\text{ m/s}^2$ and a velocity ceiling $v_{\max} = 2.00\text{ m/s}$. If the planner outputs consecutive velocity samples $\dot{q}_k = 1.95\text{ m/s}$ and $\dot{q}_{k+1} = 1.95\text{ m/s}$ separated by $T_s = 20\text{ ms}$, a cubic spline interpolator that applies an intermediate acceleration of $a_{\max}$ produces an intra-sample velocity overshoot bounded by $\Delta v \le \frac{1}{2} a_{\max} T_s$. Evaluating this bound yields:
 $$\Delta v \le \frac{1}{2} (10.0\text{ m/s}^2)(0.020\text{ s}) = 0.10\text{ m/s}$$