Consider a cylindrical continuum robot (segment) with no radial and torsional deformations, and with constant curvature along its backbone. The configuration of this system is described by the deformation coordinates (Webster and Jones, 2010):

where λ denotes the length of the backbone, κ the curvature of the segment, and ψ the angle of the robot's bending (see Figure 1A). These deformation coordinates are determined by the lengths l = [l¹, l², l³]^⊺ of the three pneumatic chambers. By using simple trigonometry, the relations between these two vectors can be obtained as follows (Sadati et al., 2017):

where r denotes the backbone-chamber distance. The continuum robot is driven by three independent pressures p = [p¹, p², p³]^⊺ ∈ P that are applied to its inner chambers (see Figure 1B). The dynamic equations of this system can be derived using Lagrangian-like methods (see e.g., Falkenhahn et al., 2015; Olguin-Diaz et al., 2018). To control the robot's shape, it is important to know what input pressures are needed to achieve a desired (final) deformation. This steady-state relation is characterized by the nonlinear mapping p=f(q):Q↦P, which can be found via the principle of virtual work (Hamill, 2014):

where U^b, U^e, U^d, and U^p denote the potential energies resulting from the body loads, external loads, elastic deformations, and pneumatic pressures, respectively. Note that the fourth term above yields a function that must be solved for p (by exploiting Equation 2) to obtain the pressure-deformation relation. However, computing the above energy terms is a difficult task that requires the exact identification of the system.

The complex properties of pneumatic continuum robots make it difficult to derive an expression relating pressures to deformations. In this work we show how an adaptive computational model can be used to approximate such nonlinear relation.

Consider first that the robot performs a series of babbling-like motions (Saegusa et al., 2009) around the workspace of interest described by P×Q. Let us assume that a set of T sampling points p(τ) and q(τ) are collected at the time instant τ and grouped into the following training data vector:

The vectors Equation(4) will be used for training a self-organizing map (SOM) (Kohonen, 2013). An important property of these maps is that they reduce the dimension of the input space into a 2D lattice while preserving its topology. Neighboring neurons represent configurations x^k that have “similar” pressure-deformation values.

The network has N computing units arranged in a 2D lattice. Each neuron has an associated weight vector w^j of the same dimension as x^k. Training is done by sequentially presenting each input pattern x^k to the network and finding the weight vector that best matches its values. This “winning” neuron satisfies:

where i denotes its index over the N lattice nodes. The best matching neuron is placed at the center of a neighborhood of cooperating nodes. Let h^ij denote the neighborhood function centered at i for an active neuron j. To make the excitation proportional to the distance from the center, a common choice is to use the Gaussian function:

where rⁱ and r^j denote the ith and jth node's 2D position within the lattice (e.g., rⁱ = [20, 15]). The variable σ_t>0 specifies the effective cooperation width, i.e., the influence that neuron i exerts over the nearby neurons. The jth weight vector is computed with the following update rule (Sakamoto et al., 2004):

for a learning gain γ_t>0. By using Equation 6, the degree of adaptation of wtj exponentially decreases with its separation from the center.

The variables σ_t and γ_t control the network's plasticity. These are first given large values to allow for coarse adaptations, and then decreased to fine tune the network's structure as follows:

for σ~,γ~>0 as the fine tuning parameters, and η >0 as its time constant.

We fabricated an elastomer-based soft manipulator (Agarwal et al., 2016; Schmitt et al., 2018) with three inner chambers that are independently controlled with pneumatic servo-valves (Festo) via an analog board (Phidgets). The robot's configuration is measured with a camera (see Figure 1C). All computations are performed in a Linux PC using OpenCV (Bradski, 2000).

In this study, we restrict our attention to the case of planar robot motions¹. To produce plane deformations, we couple the controls p² = p³ such that a virtual pressure separated by 180° from p¹ is enforced. To differentiate between left/right bendings, we use signed curvature values and define a 2D deformation vector q = [λ, sgn(ψ)κ]^⊺, where κ is measured with vision as in Navarro-Alarcon et al. (2014).

The robot first performs a series of slow bending/stretching motions (by commanding ramp pressure signals to the valves) from which T = 734 data points x^k = [p¹, p^{2, 3}, λ, sgn(ψ)κ]^⊺ are collected. This pressure/deformation data set is then used for training the network. The Supplementary Material Video S1 depicts the conducted experiments.

Note that if the network has too few neurons, the model has problems in separating distinct configurations (e.g., left and right bendings). The U-matrix is a useful method to visualize boundaries between different configurations (Ultsch and Siemon, 1990). It assigns high/low values if its weight vector is different/similar from those of Neighboring nodes. Figures 2A-C depict the U-matrices obtained by considering lattices of 10 × 10, 30 × 30, and 50 × 50 neurons, respectively. These figures show that boundaries (i.e., red stripes) start to appear with a higher number of nodes². For this study, we have selected a network with 60 × 60 neurons, which will be used for the results reported here.

Figures 2D,E depict the areas that are activated with the input pressures p¹ and p^{2, 3}. High values can be interpreted as deformations that are “mostly” enforced by either p¹ or p^{2, 3} (i.e., right bending, and left bending). The map presents little overlapping of pressure regions; this corresponds to extension/retraction motions that require coordinated actions from all inputs. Figure 2F shows the group of neurons that encode the maximum value q_max for deformations (i.e., the longest and roundest configurations), which for our case corresponds to right bendings. The identified maximum region depends on the configurations available in the data set (in our experiments, left bendings were not so prominent).

Figure 2G shows the U-matrix computed for the 60 × 60 network with symbols denoting the dominant deformation state associated with each neuron. From this data, we can create clusters to represent the most numerically distinct configurations of the robot, namely, left and right large bendings, and central positions³. The resulting deformation clusters are shown in Figure 2H (where L, C, and R denote the left, center, and right robot configurations, respectively). We can also create the pressure clusters shown in Figure 2I, where Π¹, Π², and Π³ denote areas of “mostly” p¹, “mostly” p^{2, 3}, and combined pressure actions, respectively.

The network can be used to relate a target deformation q¯ with its required pressure values p¯. By presenting a partial vector x^⊺ = [^*, ^**, q^⊺]^⊺ (for ^* as unimportant terms) to the network, we can find the weight w^j that best matches q¯, as done in Equation 5. The corresponding pressure values are simply located in the other coordinates of w^j = [w^1j, w^2j, ^*, ^*]^⊺ ≈ [p^⊺, ^*, ^*]^⊺. Using this prediction approach, our trained network showed a maximum coordinate error of (3.57%, 5.38%) for the q¯ and of (17.3%, 15.1%) for p¯.

Figures 2J–L depict examples of how the topologically preserving network can capture incremental deformations of the robot into adjacent neurons. These figures show that the end points of the “deformation trajectories” correspond to the high/low values of the performed motion. The computed maps allow the characterization of the robot's properties, including fabrication errors, unbalanced pneumatic chambers (see the slight bending in Figure 2J), or any other variation in the sensorimotor conditions.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.