TABLE 1.

Sampling of inverse statics/kinematics learning-based control for continuum robots. Intended to be exemplary, not comprehensive.

Literature	Classification criteria
	Model	Robot structure	Actuation (length)	Act. & TaskDoFs	Mapping to be learned	Samples	Task	Accuracy/mm (mean/STD/max)
Giorelli et al.	1-hidden-layer FNN
a. Giorelli et al. (2013a)	a. 21 neurons	Silicone conical robot	Tendon (310 mm/280 mm)	a. 2 & 2	$\mathit{u}\left(k\right)=f_s^{-1}\left(\mathit{p}\mathit{*}\left(k\right)\right)$	500 (8:2)	Simulation: path generation. Discrete points	2.27/1.70/9.30
b. Giorelli et al. (2013b)	b. 34 neurons			b. 3 & 3		500 (8:2)		4.2/2.8/12.3
c. Giorelli et al. (2015a)	c. 6 neurons			c. 2 & 2		405 (8:2)		22.88/11.80/59.79
d. Giorelli et al. (2015b)	d. 28 neurons			d. 3 & 3		395 (8:2)		7.35/--/22.22
Melingui et al. (2014a)	Forward: FNN. Inverse: 1-hidden-layer NN in DSL	CBHA	Pneumatic (NA)	a 3 × 2 & 3	Forward: $\mathit{p}\left(k\right)=f_s\left(\mathit{u}\left(k\right)\right)$	4,096 (7:1.5:1.5)	Comparison: robot and model postures (under the same actuation)	Inverse: 1.1 e−4 (MLP) ∼ 4.1 e−4 (RBF)/--/--
					Inverse: $\mathit{u}\left(k\right)=f_s^{-1}\left(\mathit{p}\mathit{*}\left(k\right)\right)$
Thuruthel et al.	1-hidden-layer NN.
a. Thuruthel et al. (2016a)	a. 20 neurons	a. BHA	a. Pneumatic (0.9 m)	a. 3 × 3 & 3	$\mathit{q}\left(k\right)={\widehat{f}}_s^{-1}\left(\mathit{q}\left(k-1\right),\mathit{p}\mathit{*}\left(k\right)\right)$	a. 10,000 (7:3)	Simulation: continuous path following in terms of position (P)/orientation (O)	p: <1/1.504/--
b. Thuruthel et al. (2016b)	b. 40 neurons	b. Silicone conical	b. Tendon (31 cm)	b. 12 & 6	$\text{Δ}\mathit{q}\left(k\right)={\widehat{f}}_k^{-1}\left(\mathit{q}\left(k-1\right),\text{Δ}\mathit{\theta }\mathit{*}\left(k\right)\right)$	b. 14,000 (8:2)		p: 8.5/2.8/-- O/°: 3.21/1.71/--
Chen and Lau (2016), Xu et al. (2017)	ELM, GMR, or KNNR	Silicone serpentine	Tendon (NA)	a 2 & 2	$\mathit{u}\left(k\right)=f_s^{-1}\left(\mathit{p}\mathit{*}\left(k\right)\right)$	20,000	Simulation & real robot: path following	2.1275∼ 2.5556/−/−
Lee et al. (2017a)	LWPR online update	Silicone cylindrical	Pneumatic (93 mm)	3 & 2	$\text{Δ}\mathit{u}\left(k\right)=f_k^{-1}\left(\mathit{n}\left(k-1\right),\mathit{u}\left(k-1\right),\text{Δ}\mathit{n}\mathit{*}\left(k\right)\right)$	>1,000 for initialization [FEA Lee et al. (2017b)]	Angular path following (2D) + external forces	Free space/°: 0.90/0.65/2.80
								Disturbed/°: 2.49/1.74/11.03
Ho et al. (2018)	LWPR online update	Silicone cylindrical	Pneumatic (155 mm)	3 × 2 & 3	$\text{Δ}\mathit{u}\left(k\right)=f_k^{-1}\left(\mathit{u}\left(k-1\right),\text{Δ}\mathit{p}\mathit{*}\left(k\right)\right)$	—	Path following (3D) + tip load (72% robot mass)	With load: 0.98/0.26/--
Fang et al. (2019)	LGPR online update	Silicone cylindrical	Pneumatic (67 mm)	3 & 2	$\text{Δ}\mathit{u}\left(k\right)=f_k^{-1}\left(\mathit{u}\left(k-1\right),\text{Δ}\mathit{z}\mathit{*}\left(k\right)\right)$	300	Path following (2D visual servo) + tip load	Free space/pixel: 5.4/--/11.5
Bern et al. (2020)	a. 2-hidden-layer FNN (20 × 2)	Silicone cylindrical	Tendon (20 cm)	a. 3 & 2	$\mathit{u}\left(k\right)=f_s^{-1}\left(\mathit{p}\mathit{*}\left(k\right)\right)$	a. 308 (9:1)	a. Real robot	Real robot: 6.2∼9.2/−/−
	b: 3-hidden-layer FNN (25 × 3)			b. 3 × 2 & 2		b. 15414 (8:1:1)	b. Simulation 2D path following	—

^aOnly the actuation dimensions that are related to the end-effector control are considered; “×2” or “×3” means the number of segments in the manipulator.

(F)NN, (feed-forward) neural network; GMR, Gaussian mixture regression; FEA, finite element analysis; DSL, distal supervised learning; KNNR, K-nearest neighbors regression; STD, standard deviation; (C)BHA, (compact) bionic handling assistant; LWPR, locally weighted projection regression; MLP, multilayer perceptron; ELM, extreme learning machine; LGPR, locally Gaussian process regression; RBF, radial basis function.