Soft robotics is an emerging field in the robotics sciences and enjoys increasing attention in the scientific community (Bao et al., 2018). An important part of this field is mobile soft robotics, which allows locomotion in unknown and unstructured (Katzschmann et al., 2018) as well as potentially dangerous environments (Tolley et al., 2014). In order to navigate a robot through any environment, some sort of feedback is needed. As discussed in Santina et al. (2017), high gain feedback control results in good tracking performance, but imposes a reduction in the compliance of the controlled system. Therefore, it takes away the essential characteristic and greatest advantage of a soft robot—its softness (Rus and Tolley, 2015). When it comes to soft robots, usually the dynamics of inputs are indirectly coupled with the dynamics of outputs and the coupling is time-delayed (PneuNets: pressure—angle, SMA: heat—contraction, refer to Lee et al., 2013). In order to take this into account, a cascaded control architecture has been established (see, e.g., Marchese et al., 2014; Hofer and D'Andrea, 2018). In the case of pneumatically operated robots, the inner loop controls the pressure and the outer loop controls the pressure reference (see also Figure 11B). In order to preserve softness, the feedback gain of the outer control loop needs to be low. Most of the pressure reference should therefore be generated by a feed forward term (Santina et al., 2017). There is a trend to implement the feed forward term by using Iterative Learning Control (Bristow et al., 2006; see, e.g., Santina et al., 2017; Zhang and Polygerinos, 2018; Hofer et al., 2019). As shown in Santina et al. (2020), the typical soft properties of a soft robot can also be preserved with a model-based feed forward term when doing position control.

All the soft robots discussed so far are stationary. Thus, position control refers to the position of the end effector and not to the position of the entire robot. However, the same principles are also valid for mobile soft robots. Most mobile soft robots, such as in Shepherd et al. (2011), Godage et al. (2012), Tolley et al. (2014), Qin et al. (2019), and Schiller et al. (2019), are feed forward-controlled with predefined gait patterns. In order to enable such robots to move autonomously even in unknown terrain, a locomotion controller is needed that can generate any gait path. For solving this task, different methods have been employed, such as sine generators, central pattern generators (CPG), predefined trajectories, finite state machines, or heuristic control laws (Pratt et al., 2001). An example for sine generator-based locomotion control is presented in Horvat et al. (2015, 2017) for a salamander-like robot. The method enables to operate the robot by only two drive signals, i.e., forward and rotational speed. The main contribution here is the skilful synchronization of spine and legs motion, which is very robot-specific. In Ijspeert (2008), the suitability of central pattern generators, i.e., biologically inspired neural circuits capable of producing coordinated patterns for robot's locomotion, are discussed. It is concluded that CPGs are well-suited in general and especially for distributed implementations (e.g., for snake-like or reconfigurable robots). However, there is neither a sound design methodology to solve a specific locomotor problem nor a solid theoretical foundation. In order to implement CPGs in a meaningful way, the basic gait pattern must therefore be known from the outset, which again is robot-specific. An example for the automatic generation of optimal joint-trajectories is given in Bern et al. (2019). By using a forward shooting method and an FEM-based direct kinematics simulation, high-level goals, such as forward speed or direction of movement of various soft walking robots can be met. This method does not require a priori knowledge of a motion pattern, but can not be used online without restrictions (computation time, stability, .). However, it can be well used to find robot-specific gait patterns.

Hence, for locomotion control of a robotic platform, a robot-specific motion strategy must be known. This paper analytically derives a robot-specific mapping of desired motion (forward and rotational speed) to joint coordinates for the gecko-inspired robot from Schiller et al. (2019), which is briefly described in section 2. The mapping function is referred to as “gait law” and is presented in section 3. In section 4, the direct kinematics of the robot are approximated by a polynomial by means of simulation and experiments to allow a fast evaluation. This is necessary to implement a control strategy in section 5 that maintains the softness of the robot and allows it to approach arbitrary references in the task space. The control strategy is referred to as Gait Pattern Generator. Figure 1 shows the systematic procedure of this paper. To summarize, the paper contributes in two ways: (i) it derives the robot-specific motion strategy for the gecko-inspired robot and (ii), for a given robot-specific motion strategy, it provides a method to control the robot's position. However, the underlying assumptions of the former can also be transferred to other soft robots, since the ability to adapt to the environment is exploited herein.

The soft robot this paper deals with has five limbs (four legs and a torso) and four feet that can be operated independently. Therefore, its joint space has nine dimensions: the five bending angles of the limbs α = [α₀, α₁, α₂, α₃, α₄] and the four states of fixation actuators f = [f₀, f₁, f₂, f₃]. Since its locomotion is only possible within two dimensions, its description in task space needs only three coordinates: the x and y position of the robot ^Ox and its orientation ^Oε, described in the global (Cartesian) coordinate system {O}. Thus, the task space has three dimensions. A photograph of the prototype of this robot is depicted in Figure 2A and Table 1 summarizes its specifications. In order to evaluate the performance of the robot, the test bench shown in Figure 2B was built with an embedded camera system. To measure the bending angles α, the robot orientation ε, and the robot position x, apriltags (Wang and Olson, 2016) were fixed on its body. For a more detailed description of the experimental setup, refer to the Supplementary Material.

The straight gait of the robot can be described by a single variable—the reference bending angle of the torso ᾱ2. All other variables of the joint space can then be described as a function of ᾱ2 by means of the gait law for the straight gait, which was derived in Seibel and Schiller (2018):

For a constant cycle time, the torso's bending angle is the essential measure for the forward velocity. Therefore, q₁ as the signal driving the forward velocity is introduced, and for straight gait, ᾱ2=q1 is set. In order to operate the robot with different velocities, the angle reference ᾱ(q1) for a given step size q₁ is inverted after a certain time interval t_move. Hence, it jumps from ᾱ(q1) to ᾱ(-q1). The corresponding fixation reference f must also be inverted.

The next step is to determine how the robot behaves in the task space for each point (q₁, q₂) in the velocity space—that is, how it moves per cycle and by how much it rotates. Thus, the bivariate polynomial Δx(q₁, q₂) is searched for which approximates the transformation of the velocity space into the task space (compare Figure 1). The form of the polynomial is defined as follows:

In order to identify the coefficients, the velocity space is gridded and for each set of values the motion of the robot is measured. This can either be done experimentally or the simulation model is used and the movement is simulated. The result of both approaches depends on the way they are implemented. Therefore, the influencing factors must be identified and their value must be meaningfully determined. Table 2 summarizes the conditions under which the following experiments or simulation were carried out. A detailed discussion of the experimental conditions can be found in the Supplementary Material.

Figure 7 shows the results of simulation (Figure 7A) and experiment (Figure 7B). In both cases, the velocity space was gridded with q₁ ∈ {50, 60, ⋯ , 90} and q₂ ∈ {−0.5, −0.3, ⋯0.5} and a measurement was performed for each grid point. A simplified representation of the extreme poses of the resulting gait illustrates the movement. The tip of the torso of the initial pose is always at the position (q₁, q₂) and the orientation of the robot faces upwards. The resulting translation [ΔxΔy]^⊤ per cycle is indicated by a red arrow. Besides, the orientation of the robot after a cycle is represented by a green line. The heat map in the background shows the resulting rotation Δε(q₁, q₂) per cycle. The numerical value of this function is noted in a green box below the individual measurements. In the figure of the experiment (Figure 7B), the standard deviation of the translation is shown as a red ellipse with the corresponding semi axes. The standard deviation of the rotation is visualized as a light green triangle with an opening angle of 2std(δε). The blue arrow shows the polynomial fit of the translation and the blue line the polynomial fit of the rotation at the corresponding grid point. A detailed view of a single experiment is shown in Figure 8.

In contrast to the experiment in section 3.2, in Figure 7, a clear deviation between simulated and experimental results can be observed. The resulting rotation and the shift in transverse direction are noticeably higher for all grid points. The simulation model does not reproduce friction effects or external disturbances, such as the influence of the supply tubes. In the previous experiment (from section 3.2), these effects played a subordinate role because of the reduced speed and the relatively short distance traveled. This experiment was executed at full speed (t_move = 1 s); thus, friction has a significantly larger influence. Furthermore, we can observe that the experiment is not symmetrical, meaning that swapping the sign of q₂ does not yield to mirrored behavior [Δx(q₁, q₂)≁−Δx(q₁, −q₂)]. This can be attributed to manufacturing inaccuracies of the robot and an optimizable pressure-bending angle calibration. The calibration procedure and associated difficulties are also discussed in the Supplementary Material. A final observation is that, in the experiment, the resulting rotation decreases for a large step length q₁. This is different to the simulation, where the resulting rotation increases steadily with increasing step length. For large values of q₁ and q₂, the gait law prescribes relatively large reference angles. If these are out of range of calibration of the respective actuator, the reference pressure is saturated to prevent damage to the robot. Exactly this effect occurs in the upper part (q1≥80∘) of Figure 7B. Therefore, the poses here deviate much more from their simulated counterparts in Figure 7A than in the lower part of the figure (q1<80∘). Apart from the “over-simulation” and the missing saturation effect, the simulation reproduces the behavior very well. It can be seen as the behavior of a robot that has been perfectly manufactured and calibrated, consisting of actuators as robust as saturation is no longer necessary, whose feet have the optimum torsional stiffness, and where all friction effects have been reduced to a minimum. For the implementation of the Gait Pattern Generator, however, the actual interest focuses on the polynomial fit of the motion. In most cases, the second-order fit shown in blue matches the measurement or is at least within the standard deviation. The coefficients for the polynomial Δx(q₁, q₂) for Equation (16) are listed in Table 3.

The last step to control the robot's position is the calculation of the optimal tuple q^* to move from the current position x closer to a given target position x̄ (compare Figure 1).

The aim of this work was position control of the gecko-inspired soft robot from Schiller et al. (2019) in Cartesian space. The solution to this complex task is based on two major simplifications: (i) the formulation of a gait law to reduce the state space of the robot from nine to two dimensions and (ii) the approximation of the direct kinematics to allow a fast evaluation. The gait law restricts the choice of possible references extremely; e.g., only specific trotting gaits are allowed. In this work, it was successfully examined whether a position control system can function with this limitations. However, it has not been investigated whether a larger permitted choice of references leads to better results. In fact, it is possible that the introduction of additional generalized coordinates or a different gait law may lead to a better performance of the robot. Furthermore, neither frictional effects nor any dynamics were considered. Also, by approximating the direct kinematics in the polynomial Δx, an assumption is made which is fulfilled only in very few cases (compare section 5.2). Instead of using the approximation, the simulation model could also be employed to find the best possible reference for the current situation. But the simulation of one step takes an average of 0.1 s on an AM335x 1GHz ARMⓇ Cortex-A8 processor, which is used for control. With an average of 10 evaluations of the direct kinematics required to find the reference leading to the minimum distance, this adds up to 1 s. In contrast to a polynomial approximation where the Jacobi matrix can be easily formed to find the minimum efficiently, no analytical Jacobi matrix has been formulated for the simulation model so far. This means that when the simulation model is used, calculation would require most of the time of the cycle. However, the experiments show that the robot always reaches the target, even if the assumptions made in the derivation of the Gait Pattern Generator are not fulfilled and the approximation of the direct kinematics was done for a robot of different dimensions.

The path planning algorithm implemented is very basic, as it minimizes the Euclidean norm of the target vector, i.e., it dictates the direct path from the current position to the target. The gait law provides an intuitive way (forward and rotational speed) to control a quite complex robot and the approximation of the direct kinematics provides the resulting quantitative motion. This opens an interface to a wide variety of more dedicated path planning algorithms, as the robot can now be treated as a unicycle. For example, the path could be planned using Cartesian polynomials (Siciliano et al., 2010) and thus the robot orientation could also be controlled. Although the softness of the robot is very complex to model, it also allows the formulation of very drastic references, even if these cannot be fulfilled at all, as hindered by the closed kinematic chain. How these contradictory demands are solved is then “computed” by the body itself. Conventional parallel kinematic robots, such as the Stewart-Gough platform, would be damaged in this case. The gecko-inspired soft robot is therefore a good example of Embodied Intelligence (Cangelosi et al., 2015) or Morphological Computation (Pfeifer and Gómez, 2009) since it does the right thing “intuitively.” This is in agreement with the principle of controlling soft robots mainly in a feed forward way in order to maintain and make use of their softness (Santina et al., 2017). The cascaded controller structure, as discussed in the introduction, can therefore also be applied to position control of mobile robots. The method of deriving a basic locomotion strategy like the presented gait law by very simple (feet rotate only in swing phase), but mathematically (with the constant curvature model) unfulfillable assumptions, can be transferred to any other soft mobile robot. Although this needs to be done individually for each robotic platform, this work can serve as a reference for future and/or existing robots.

The raw data generated for this study are available on request.

LS derived the Gait Pattern Generator, performed the experiments, and discussed the results. LS and AS wrote and revised the manuscript. AS and JS supervised the project. All authors contributed to the article and approved the submitted version.

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.