Tendon-driven continuum robots can reach through constrained spaces, but contact along their bodies is invisible to the sensors at their motors. We show that this is a structural limitation rather than a tuning problem: the map from contact forces to tendon measurements has a null space of dimension at least seven, so no estimator that leaves the contact support free can locate contact. When the obstacle geometry is known, however, non-penetration and complementarity make the contact force locally identifiable, and we prove that the resulting system with unknown contact inputs is locally exponentially strongly detectable. We present a constrained moving horizon estimator (MHE) that encodes these conditions and runs in real time as one quadratic program per sample, and we give a local, mode-wise robust stability result for it. In a finite- element benchmark of a two-section robot pressing and sliding on a wall, the estimator located contact to 0.9 mm and estimated contact force to 0.22 N, compared with 65 mm and 0.72 N for a tuned extended Kalman filter (EKF) with state projection. The estimator tolerated millimeter- level errors in the obstacle position and unmodeled wall friction, under which the projected EKF failed. Under randomized model mismatch over 40 trials, including unmodeled tendon friction, it remained better in force and location in every trial. Without contact, the MHE offered no advantage over the EKF, which confirms that its benefit comes from the contact constraints. This report extends a letter submitted to IEEE Control Systems Letters with complete proofs and additional experiments.
Figures & tables
Work
Sensing
Estimates
Estimator
Xu & Simaan [ 2 ]
actuation forces, configuration
wrench
statics
Bajo & Simaan [ 3 ]
segment kinematics
contact, segment
thresholds
Aloi et al. [ 4 ]
3D shape, tensions
loads and locations
EKF
Ferguson et al. [ 5 ]
position or strain
shape and loads
batch estimation
Johnston et al. [ 6 ]
shape, motor current
wrench
momentum observer
Prakash et al. [ 7 ]
strain, tip pose, tensions
multiple contact forces
factor graph
Table 1: Contact estimation for continuum robots: sensing and estimated quantities.
Figure 1: Problem setting. (a) Finite-element ground truth during contact (run 1, side view), with the 13 candidate contact points and the true contact force λ at the tip. (b) Singular values of the sensitivity G from contact forces to tendon measurements (median and 10th–90th percentiles over 990 configurations). Seven directions lie in kerG and only approximately three exceed the noise level.
Figure 2: Identifiability along five contact runs. (a) Singular values of G . (b) A force distribution in kerG , which changes no tendon length. (c) Similarity of neighboring columns of G : adjacent contact points look almost identical to the tendons. (d) Cramér–Rao bound on the total contact force with a known active set, compared with the error of the real-time MHE; the gap of approximately 10 × is due to model mismatch rather than sensor noise.
Shape
Force
Location
Spurious
Penetr.
Detect.
Time
Estimator
(mm)
(N)
(mm)
(N)
(mm)
(%)
(ms)
PCC
13.8 (2.3)
–
–
–
–
–
0.5
EKF, no contact
23.0 (3.2)
–
–
–
–
–
91
EKF + forces
5.6 (0.4)
2.44 (0.62)
–
6.6
25.8
41
137
EKF + projection
3.2 (0.3)
0.72 (0.26)
65 (12)
0.50
9.0
95
140
MHE, IPOPT
3.0 (0.3)
0.25 (0.12)
2.1 (0.6)
0.009
0
98
580
Table 2: Mean (standard deviation) over five contact runs. Location: error in the force-weighted contact arc length. Spurious: mean force estimated at points more than 10 mm from the wall. Penetr.: maximum estimated penetration into the wall. Detect.: fraction of samples with correct contact detection (threshold 0.3 N).
Figure 3: Contact estimation over time (run 1). (a) Total contact force. (b) Force-weighted contact location along the arc length, shown when the force exceeds 0.3 N.
Figure 4: Nominal comparison over five contact runs. Top: shape error, total contact force RMSE (log scale), contact location error, and maximum wall penetration for each estimator. Bottom: total contact force and contact location over time for run 1.
Real-time MHE
EKF + projection
Condition
Force
Location
Shape
Force
Location
Shape
Nominal
0.22
0.9
3.0
0.72
65
3.2
Wall assumed 2 mm too close
0.22
1.0
3.1
1.16
118
3.5
Wall assumed 2 mm too far
0.24
0.9
3.2
0.50
36
3.2
Wall assumed ±5 mm off
≤ 0.27
≤ 1.0
≤ 3.8
–
–
–
Wall friction μ=0.1
0.61
1.3
22.4
19.4
286
28.7
Table 3: Robustness with unchanged tuning (five runs per condition): force RMSE in N, location and shape error in mm.
Figure 5: Robustness. (a) Force RMSE versus error in the assumed wall position. (b) Normal-force RMSE under unmodeled Coulomb friction at the wall. (c) Mid-body contact.
Figure 6: Convergence of the real-time MHE. (a) Shape and (b) total-force deviation from the unperturbed estimate after eight large perturbations of the estimation window. (c) Force RMSE and shape error versus tendon-length noise.
Metric
Real-time MHE
EKF + projection
MHE better
Wilcoxon p
Force RMSE (N)
0.40 [0.31, 0.51]
4.8 [3.7, 6.1]
40/40
2×10−12
Location error (mm)
2.8 [2.0, 3.6]
193 [155, 232]
40/40
2×10−12
Shape error (mm)
7.5 [6.6, 8.4]
10.7 [9.6, 12.0]
30/40
7×10−6
Max penetration (mm)
0.37 [0.15, 0.65]
15.6 [10.4, 22.1]
37/40
3×10−11
Contact detection (%)
94
83
–
–
Table 4: Model-mismatch Monte Carlo (40 trials): mean and 95% bootstrap confidence interval.
Figure 7: Model-mismatch Monte Carlo (20 runs, two noise realizations each). (a) Force RMSE. (b) Contact location error (values below 0.1 mm shown at 0.1 mm). (c) Shape error. (d) Real-time MHE force RMSE versus true stiffness, colored by tendon friction.
Nominal
Heavy payload
Outliers
Estimator
Shape
Payload
Shape
Payload
Shape
Payload
PCC
4.14 (0.39)
–
9.42 (1.46)
–
5.24 (0.40)
–
EKF
3.52 (0.37)
8.3
7.70 (2.08)
20.4
4.01 (0.37)
10.2
MHE
3.55 (0.38)
8.3
7.75 (2.10)
20.1
3.98 (0.39)
10.8
Robust MHE
3.55 (0.39)
8.6
7.89 (2.09)
22.4
3.65 (0.37)
9.0
Table 5: Motion without contact (five runs per condition): mean backbone shape error in mm (standard deviation) and payload RMSE in g.