Improving Robot Position Tracking Accuracy: Acceleration Feedforward Optimization Journey

The existing numerical differentiation approach to acceleration feedforward (FF) achieved only about 5% improvement. This was far below theoretical expectations.

The key question: "How much better would it be if we used the motion planner's acceleration directly?"

To answer this question, we systematically compared 6 acceleration acquisition methods.

Conclusion First

Method	Position RMSE	Improvement	Torque RMSE
A0 (None)	0.001389 rad	Baseline	2.77 Nm
A1 (NumDiff)	0.001141 rad	+17.8%	4.29 Nm (+55%)
A4 (LPF)	0.001095 rad	+21.2%	2.68 Nm (-3%)

Optimal Method: Planner acceleration + Low-Pass Filter (alpha=0.3)

• Position accuracy 21.2% improvement
• Torque variation 3% reduction (compared to baseline)

System Architecture

Key Finding: ROS2 JointTrajectoryController was already computing acceleration through Quintic Spline interpolation. No separate numerical differentiation is needed.

New Problem: Quintic Spline guarantees continuity of position/velocity/acceleration, but Jerk (derivative of acceleration) is discontinuous. High-frequency components occur at waypoint boundaries, potentially causing mechanical vibration.

Robot Dynamics Equation

$$\tau = M(q)\ddot{q} + C(q, \dot{q})\dot{q} + g(q) + \tau_{friction}$$

Term	Description	Relationship to FF
$M(q)\ddot{q}$	Inertia torque	Acceleration FF target
$C(q, \dot{q})\dot{q}$	Coriolis/centrifugal torque	-
$g(q)$	Gravity compensation torque	-
$\tau_{friction}$	Friction compensation torque	-

Comparison of 6 Acceleration Acquisition Methods

Code	Method	Description
A0	None	No acceleration FF (baseline)
A1	NumDiff	Numerical differentiation of velocity command (existing method)
A2	ABG	Alpha-Beta-Gamma filter
A3	Planner	Direct use of JTC acceleration (no filter)
A4	LPF	JTC acceleration + Low-Pass Filter
A5	JerkLimit	JTC acceleration + Jerk limiting

Problems with Numerical Differentiation (A1)

acceleration[k] = (velocity[k] - velocity[k-1]) / dt;

Problem	Cause	Result
Phase delay	Discrete differentiation uses only current and past samples	Output delayed from actual acceleration
Noise amplification	Differentiation amplifies high-frequency components	55% increase in torque variation

Low-Pass Filter (A4)

Using planner acceleration directly (A3) causes Jerk discontinuity at waypoint boundaries. LPF removes these high-frequency components to create a smooth acceleration signal.

y[k] = alpha * x[k] + (1 - alpha) * y[k-1];

In other words, new output = (alpha x new input) + ((1-alpha) x previous output).

Variable	Meaning
x[k]	New incoming acceleration value
y[k-1]	Previous filter output
alpha	New value reflection ratio (0-1)

With alpha=0.3, the output is 30% new value + 70% previous value, smoothing sudden changes.

Relationship Between alpha and Cutoff Frequency ($f_s$ = sampling frequency 1kHz, $f_c$ = cutoff frequency):

$$f_c \approx \frac{\alpha \cdot f_s}{2\pi}$$

Frequencies below $f_c$ pass through, while those above are attenuated.

Alpha	Cutoff Frequency	Characteristics
0.1	~16 Hz	Excessive filtering, response delay
0.2	~32 Hz	Stable, slight response delay
0.3	~48 Hz	Optimal balance point
0.5	~80 Hz	Fast response, torque increase starts
0.7	~112 Hz	Weak filtering, 11% torque increase

Experimental Results

Experiment 1: Comparison of 6 Methods

Position RMSE (left) and Torque RMSE (right) comparison. A4 (LPF) is the optimal balance point

Position RMSE Ranking:

Rank	Method	RMSE (rad)	Improvement
1	A4 (LPF)	0.001190	+16.0%
2	A2 (ABG)	0.001212	+14.4%
3	A5 (Jerk)	0.001231	+13.1%
4	A1 (NumDiff)	0.001297	+8.4%
5	A3 (Planner)	0.001302	+8.1%
6	A0 (None)	0.001416	Baseline

Torque RMSE:

Method	Torque RMSE	Evaluation
A5 (Jerk)	2.61 Nm	Stable
A4 (LPF)	2.63 Nm	Stable
A3 (Planner)	2.64 Nm	Stable
A0 (None)	2.76 Nm	Baseline
A1 (NumDiff)	4.15 Nm	+50% (Warning)
A2 (ABG)	5.19 Nm	+88% (Eliminated)

Reason A2 (ABG) was eliminated: Good position but torque variation too high.

Experiment 2: LPF Parameter Tuning

Position RMSE vs Torque RMSE tradeoff by alpha value. alpha=0.3 selected

Alpha	Cutoff Frequency	Position RMSE	vs alpha=0.2	Torque RMSE
0.1	~16 Hz	0.001227 rad	+1.0%	2.85 Nm
0.2	~32 Hz	0.001240 rad	Baseline	2.69 Nm
0.3	~48 Hz	0.001118 rad	-9.8%	2.76 Nm
0.5	~80 Hz	0.001150 rad	-7.3%	2.97 Nm
0.7	~112 Hz	0.001071 rad	-13.6%	2.99 Nm

Reasons for Selecting alpha=0.3:

• Position improvement: 9.8% improvement compared to alpha=0.2
• Torque stability: 2.76 Nm, nearly identical to baseline (2.77 Nm)
• Safety margin: alpha=0.7 has 11.2% torque increase

Experiment 3: Final Benchmark

Final comparison of A0 (baseline), A1 (numerical differentiation), A4 (LPF)

Position Tracking Performance:

Method	Position RMSE	Std Dev	Improvement
A4 (LPF)	0.001095 rad	0.000087	+21.2%
A1 (NumDiff)	0.001141 rad	0.000064	+17.8%
A0 (None)	0.001389 rad	0.000071	Baseline

Torque Stability:

Method	Torque RMSE	Std Dev	Evaluation
A4 (LPF)	2.68 Nm	0.37	Stable (-3%)
A0 (None)	2.77 Nm	0.52	Baseline
A1 (NumDiff)	4.29 Nm	1.04	Unstable (+55%)

Why Planner Acceleration Alone (A3) Is Insufficient

Using planner acceleration directly without filtering achieves only 8.1% improvement (half of A4).

Cause: Jerk discontinuity at waypoint boundaries generates high-frequency components that burden the controller.

Why LPF (A4) Is Better Than Jerk Limiting (A5)

Aspect	LPF	Jerk Limiting
Response characteristic	Exponential decay	Linear ramp
Convergence speed	Fast initial response, gradual convergence	Constant rate approach, slow convergence
Example: delta_acc=20 rad/s^2, MAX_JERK=50 rad/s^3	Fast	400ms required

Warm-up Effect

We discovered a warm-up effect during experiments:

Condition	Before Warm-up	After Warm-up	Difference
A3 (Planner)	0.001532 rad	0.001302 rad	15% improvement

Implication: Performance can degrade by 15% on cold start. This should be considered in actual robot operation.

Experimental Conditions

Parameter	Value	Reason
Test joint	J1 (shoulder)	Maximum friction (Fc=35Nm) maximizes FF effect
Velocity scale	0.3, 0.4, 0.5	Differences clearer at high speeds
Acceleration scale	0.5	High acceleration condition
Repetitions	4 per speed	Direction balance (2 forward + 2 reverse)
Trajectory	+/-30 deg toggle	Standard test motion

Key Takeaways

1. Limitations of existing numerical differentiation (A1): Phase delay and noise amplification from discrete differentiation increases torque variation by 55%.

2. ROS2 JTC already computes acceleration. Planner acceleration can be directly utilized without separate numerical differentiation.

3. Jerk discontinuity problem: Using planner acceleration without filtering degrades performance due to high-frequency components at waypoint boundaries.

4. Optimal method: LPF (alpha=0.3)

   • Position RMSE: 21.2% improvement
   • Torque RMSE: 3% reduction
   • Simultaneously improves position accuracy and torque stability

5. Warm-up effect: 15% performance degradation may occur on cold start.

A simple first-order Low-Pass Filter was the optimal solution that improves both position accuracy and torque stability simultaneously.