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Abstract

Although soft robots show safer interactions with their environment than traditional robots, soft mechanisms and actuators still have significant potential for damage or degradation particularly during unmodeled contact. This article introduces a feedback strategy for safe soft actuator operation during control of a soft robot. To do so, a supervisory controller monitors actuator state and dynamically saturates control inputs to avoid conditions that could lead to physical damage. We prove that, under certain conditions, the supervisory controller is stable and verifiably safe. We then demonstrate completely onboard operation of the supervisory controller using a soft thermally actuated robot limb with embedded shape memory alloy actuators and sensing. Tests performed with the supervisor verify its theoretical properties and show stabilization of the robot limb's pose in free space. Finally, experiments show that our approach prevents overheating during contact, including environmental constraints and human touch, or when infeasible motions are commanded. This supervisory controller, and its ability to be executed with completely onboard sensing, has the potential to make soft robot actuators reliable enough for practical use.

Introduction

One of the most prevalent claims about soft robots is their intrinsic safety when interacting with humans or the environment.^1,2 Less commonly discussed are new challenges in safety introduced by the novel soft actuators³ required for generating motion. For rigid robots, typical electromagnetic actuators (motors) are of little concern in comparison to the robot body's inertia in damaging its surroundings or causing injury.^4,5 In contrast, soft actuators can fail dramatically, as practitioners may recognize. Informally, pneumatic balloons can pop,⁶ thermal actuators can overheat and cause fire risks or burns to human skin,⁷ and dielectrics can cause dangerous arcing,^8,9 among others. As of yet, these risks have been mitigated by simple bespoke system designs, hard limits on actuation input,¹⁰ or open-loop actuation.^11,12 Incorporating automatic control into soft robots demands more generalizable and robust approaches to actuator safety.

This article proposes a feedback control framework that ensures safety of a class of soft robot actuators. The framework employs a model-based supervisor that works in tandem with a user's arbitrary nominal controller (Fig. 1e–g). We demonstrate our framework on a thermal shape memory alloy (SMA) actuator with two different nominal controllers in the presence of environmental contact (Fig. 1a–d). This task presents a generalizable challenge as the cause of failure, excess heat, can only be indirectly monitored and controlled.

FIG. 1.

When feedback controllers for soft robots encounter unmodeled physical interactions, such as with the environment or humans, the resulting contact loads or kinematic constraints can cause damage to the robots themselves (a). Our approach prevents these unsafe situations—in our case, our supervisory controller prevents overheating (b, c). The example testbed is a soft limb with a single degree of freedom (d). Our feedback framework (e) assumes separate actuator dynamics versus body dynamics (f). The supervisor operates in conjunction with any arbitrary nominal controller to apply safe inputs (g).

Specifically, this article contributes: 1.

A provably safe, provably stable supervisory controller for soft robot actuators modeled as affine systems.

A provably safe integration of the supervisor with any underlying nominal controller.

A verification of the framework on a soft robot limb, maintaining safe actuator states, in an otherwise dangerous task.

Background: robot safety

Reliable use of robots in practical settings requires maintaining safe operation and consistent performance throughout their life span, regardless of environmental contact or human interaction.^4,5 The informal concept of “safety” as limiting force or position^13–15 is well suited for soft robots since mechanical conformability to contacting surfaces naturally restricts motions.¹⁶ Even when such limits are exceeded, some soft polymers can self-heal when mechanically damaged and recover their material properties.^6,9 However, most soft robots rely on mechanism design for safety,^17–20 and there are only few examples of computational intelligence for verifiable behaviors.²¹

In contrast, a formal specification of “safety” is a set inclusion problem, where if a robot's state remains within a certain set for all time, then it is considered safe: the set is invariant under the system's dynamics.²² Computational techniques such as control barrier functions,²³ model-predictive control,²⁴ and formal methods²⁵ use this framework for safety verification. However, each approach comes with computation and implementation challenges, particularly the requirement of an accurate low-dimensional dynamics model, which is a longstanding challenge in soft robotics.

Soft actuator safety and degradation

The unique material properties of actuators used in soft robotics introduce additional safety challenges: catastrophic pressure failures, high temperatures and fire, or high-voltage arcing. As of yet, these dangers have been indirectly addressed by reducing actuation force²⁶ or by introducing design modifications.²⁷ Feedback control has also only indirectly addressed actuator safety, using approaches such as low impedance²⁸ without verification, open-loop planning^29,30 only in known environments, or optimization-based control with only state constraints.³¹

For thermal actuators in particular, prior work in safety has focused on sensing intrinsic actuator states (i.e., temperature) via inverting a constitutive model^32,33 and applying a fixed threshold,^10,34–36 but these have not shown environmental contact⁷ or formal verification. In addition to physical safety, SMA actuators in particular are known to suffer from degradation due to thermal and mechanical cycling,^17,37–39 which when viewed as a temperature constraint⁴⁰ can also be formulated as a safe control problem.

Approach and applicability

Our framework considers dynamic saturation as a form of supervisory control, motivated by prior work that uses reachability computations to determine “activation” of a supervisor.⁴¹ The proposed framework is a simplistic version of the formal supervisory controller framework^42–44 with only a single switching state.

Although our proposed approach makes a number of assumptions about the robot's actuator dynamics, as are relevant to the use of SMA wires in our application case, it may be generalizable among a wider class of soft actuators. This article assumes one internal state and one control input per actuator, that the actuator states are independent, and that actuator dynamics are an affine system.³⁰ Other soft actuators may also be modeled by a single parameter, such as piston displacement in pneumatic or hydraulic actuators,^45,46 or cable retraction for cable-driven soft robots.^47–49 All these soft actuators are also monotonic control systems, in that state varies monotonically with input, a key insight for our safety verification. And, many soft actuation methods obey linear or affine dynamics, including motors for cables⁴⁷ or internal state for twisted-and-coiled actuators.⁵⁰ For nonlinear soft actuators, our supervisor may be applied via local linearizations, which produces affine dynamics when calculated around nonequilibrium points.³¹ This article includes a study of our controller's tuning parameter to assist in its application conservatively to such linearizations.

Supervisory Control for a Soft Robotic Actuator

To derive our supervisory controller, we first formulate the model of our system and derive a simple but low-performance static input bound for safety. We then address these limitations by presenting a dynamic saturation condition, combining that saturation with a nominal controller, and finally verifying safety of the composed closed-loop system.

System model

The physical states of our representative soft robot, powered by SMAs, include the robot body's bending curvature and the actuators temperatures, described later in the Robot and Actuator Model and Calibration section. For a formulation that is generally applicable to soft actuators of all kinds, we abstract these states into $x \in ℛ^{n}$ , consisting of the safety-critical actuator states $w \in ℛ^{m}$ and non-safety-critical states $z \in ℛ^{(n - m)}$ relating to the remainder of the system, as in . The system's inputs are $u \in ℛ^{p}$ . We assume dynamics in the form $x (k + 1) = f (x (k), u (k)) = [\begin{matrix} g (z, w) \\ h (w, u) \end{matrix}],$ (1)

where the safety-critical actuator states influence the non-safety-critical states via $g (z, w)$ but not vice versa. We again suggest that local linearization of coupled actuator-pose systems may enable equations of this form.

Our supervisory control system does not require knowledge of the non-safety-critical dynamics $g (z, w)$ , presumably those related to the robot's body deflection or pose, so no soft beam mechanics models are required here. We do assume that the actuator dynamics $h (w, u)$ are known, that our soft robot has one input per actuator state ( $p = m$ ), and actuators that are uncoupled: $w_{i} (k + 1) = h_{i} (w_{i} (k), u_{i} (k)) \forall i = 1 \dots m .$ (2)

In addition, we assume that each actuator dynamics function $h (\cdot, \cdot)$ is a linear (affine) system of the form $h_{i} (w_{i} (k), u_{i} (k)) = a_{1, i} w_{i} (k) + a_{2, i} u_{i} (k) + a_{3, i},$ (3)

as motivated by local linearizations around nonequilibrium points, which produce affine dynamics.³¹ These affine differential equations also arise from heating of thermal actuators.³⁰

Dropping the i indexing, we consider each actuator's dynamics individually, as verified safety per-actuator will then verify the whole system. We can use the affine augmentation to rewrite Equation (3) as $\tilde{w} (k + 1) = A \tilde{w} (k) + B u (k)$ (4)

where $A = [\begin{matrix} a_{1} & a_{3} \\ 0 & 1 \end{matrix}], B = [\begin{matrix} a_{2} \\ 0 \end{matrix}] .$ (5)

The dynamics for each actuator, Equation (4), are a linear single-input system, and we therefore can use linear system control techniques for the actuator itself despite the presumably nonlinear body dynamics $g (z, w)$ .

Static bounds on control input are impractical for safety

Our problem statement considers a safety-critical constraint on the actuator state of the form $w \leq w^{M A X}$ , that is, a maximum operating limit. In the augmented form, this limit is . We formally define safety as maintaining the actuator state below this limit, $\tilde{w} (k) \leq {\tilde{w}}^{M A X} \forall k$ .

From the physical intuition of our actuator, one concept for meeting this safety specification is a simple bound on the control input. Mathematically, applying an upper bound of a fixed static input of $u (k) = ū \in ℛ$ gives our closed-loop actuator dynamics as $\tilde{w} (k + 1) = A \tilde{w} (k) + B ū$ . Under mild stability assumptions, our actuator states would converge to an equilibrium point ${\tilde{w}}^{e q}$ determined by the magnitude of $ū$ . Setting $\tilde{w} (k + 1) = \tilde{w} (k) = {\tilde{w}}^{e q}$ ,

Observe next that the actuator dynamics in Equation (3) is a monotone control system,⁵¹ that is, for two different inputs u₁ and u₂ applied at the same known state w, dropping time index for brevity, $\begin{matrix} u_{1} \leq u_{2} \Rightarrow h (w, u_{1}) \leq h (w, u_{2}), \\ \Rightarrow w_{1} (k + 1) \leq w_{2} (k + 1) . \end{matrix}$ (7)

In other words, our actuator's state has a lower value if we apply less input: less electrical power applied to our thermal muscles means lower temperature. Section 1 in the Supplementary Information S1 formally shows monotonicity.

An initial concept for safety via saturation might therefore choose ${\tilde{w}}^{e q} = {\tilde{w}}^{M A X}$ and calculate the corresponding $ū$ via Equation (6). If the robot was attempting to perform a nominal task under a nominal control signal $v (k)$ , then a static bound via $u (k) = m i n (v (k), ū)$ would take advantage of monotonicity to prevent $\tilde{w}$ from exceeding ${\tilde{w}}^{M A X}$ .

Our initial attempt using this method underperformed so dramatically that it is not reported here. This static bound is impractical for multiple intertwined reasons.

First, the static bound value of $ū$ is extremely conservative in practice. In our tests, the static supervisor almost always engaged immediately upon attempting any $v (k)$ . For comparison, the results of our improved method described below were deployed in hardware in the Supervisory Control Results section and allowed inputs approximately seven times larger than this static bound, at ∼ $60 %$ duty cycle input voltage to our SMAs versus $ū = 8.3 %$ duty cycle.

Second, this static bound is open loop and therefore relies entirely on the accuracy of calibration and model fidelity in the $A$ and $B$ matrices. Soft robot models, particularly when simplified as ours, are notoriously imprecise, and therefore, it is unlikely that $u (k) = ū$ will cause convergence or safety for exactly ${\tilde{w}}^{M A X}$ .

Finally, since $u (k) = m i n (v (k), ū)$ is not necessarily differentiable at $u = ū$ , significant chatter can occur. Difficulty in analysis of the closed-loop system arises even though our control synthesis is simple.

Consequently, we seek a supervisor that balances safety with completion of a nominal task if possible, uses feedback to increase robustness, and has more favorable analysis properties. To do so, we make the important observation that the input magnitude $u (k)$ to reach some setpoint at future timesteps will vary based on the current state—the supervisor can (and should) be dynamic, not static.

The supervisor's dynamically saturating controller

To derive a dynamic saturation condition, consider what input magnitude it would take to reach an arbitrary setpoint ${\tilde{w}}^{S E T}$ at the next timestep given an observed state $\tilde{w} (k)$ , that is, $\tilde{w} (k + 1) = {\tilde{w}}^{S E T} .$ (8)

From linear systems theory, we have the celebrated result⁵² that if ${\tilde{w}}^{S E T}$ is reachable from $\tilde{w} (k)$ , it can be steered there in T-many steps with minimum-energy cost by applying a sequence of inputs according to timestep $t = 0 \dots (T - 1)$ in the horizon:

where $W_{T}$ is the controllability Grammian for discrete-time systems:

and is the pseudoinverse.

For the aggressive case of a single-step horizon where $T = 1$ , as implied by Equation (8), substituting in the definition of the Grammian from Equation (10) at the single timestep of $t = 0$ becomes

Crucially, just as with the static supervisor, monotonicity of this system gives that theoretically $u (k) < u^{*} (k) \Rightarrow \tilde{w} (k + 1) < {\tilde{w}}^{S E T}$ . A concept, then, to bring our system as close to ${\tilde{w}}^{S E T}$ as possible without exceeding ${\tilde{w}}^{S E T}$ would be to saturate by this time-varying $u^{*} (k)$ .

However, this bound also suffers when the dynamics model is imprecise. In contrast to the static supervisor, $u (k) < u^{*} (k)$ could still readily produce $\tilde{w} (k + 1) > {\tilde{w}}^{S E T}$ in practice, as the aggressive predictions rely on a perfectly known $A$ and $B$ . To allow a more conservative supervisor, we propose a scalar multiplier $γ \in (0, 1)$ to limit the input to a fraction of $u^{*} (k)$ . Our candidate dynamic supervisor's saturation limit is then

To analyze this controller's performance, we close the loop by applying Equation (12) to Equation (4), producing the autonomous dynamics of $\tilde{w} (k + 1) = (1 - γ) A \tilde{w} (k) + γ {\tilde{w}}^{S E T} .$ (13)

Make the following substitution to analyze this system as a linear system, eliminating the setpoint offset: $\tilde{w}' : = \tilde{w} - {(I - (1 - γ) A)}^{- 1} γ {\tilde{w}}^{S E T} .$ (14)

Substitution into Equation (13) gives $\tilde{w}' (k + 1) = (1 - γ) A \tilde{w}' (k) .$ (15)

The equilibrium point under consideration for the closed-loop system of Equation (15) is therefore $\tilde{w}' = 0 \Rightarrow {\tilde{w}}^{e q} = {(I - (1 - γ) A)}^{- 1} γ {\tilde{w}}^{S E T} .$ (16)

Notice this equilibrium point is not equal to our setpoint ( ${\tilde{w}}^{S E T}$ ). The inclusion of $γ$ unfortunately scales the equilibrium point in addition to slowing convergence. To resolve this last issue, we can select our ${\tilde{w}}^{S E T}$ such that our equilibrium point becomes the constraint boundary. Letting ${\tilde{w}}^{e q} = {\tilde{w}}^{M A X}$ and solving for ${\tilde{w}}^{S E T}$ , ${\tilde{w}}^{M A X} = {(I - (1 - γ) A)}^{- 1} γ {\tilde{w}}^{S E T},$ (17)

Combining Equations (12) and (18), the full form of our supervisor's dynamic saturation bound that takes the actuator state to the boundary of its constraint with a convergence rate of $γ$ is

Finally, we confirm the stability of our system under the feedback controller $u (k) = u^{M A X} (k)$ . Considering the error dynamics and performing some algebra on Equations (15)–(18), the closed-loop response is $e : = \tilde{w} - {\tilde{w}}^{M A X} \Rightarrow e (k + 1) = (1 - γ) A e (k) .$ (20)

This is the same form as Equation (15) with a change of variable, so the closed-loop system is stable if the open-loop system is stable and $γ \in (0, 1)$ . See Section 2.1 in the Supplementary Information S1 for a full proof.

Safety verification of the supervisor's controller

Operating the supervisor's controller gives the dynamics in Equation (20). We assume that the closed-loop system has been designed via the criteria in the Supplementary Information S1 to be globally asymptotically stable, ${lim}_{k \to \infty} \tilde{w} (k) = {\tilde{w}}^{M A X}$ . However, even exponential stability does not necessarily guarantee that: $\tilde{w} (k) \leq {\tilde{w}}^{M A X} \forall k \in N^{+} .$ (21)

For example, this safety condition would not hold for an underdamped single-input, single-output (SISO) linear system.

Verifying the condition (21) can be done instead by calculating an invariant set for a given constraint.^53,54 First, we pose the inequalities of the safety constraint as a polytope in the space of our autonomous system, that is, the error dynamics. Using the definition in Equation (20) for the error, $\tilde{w} = {\tilde{w}}^{M A X} \Leftrightarrow e = 0$ . The upper bound condition in Equation (21) is therefore $e \leq 0$ . Then, to perform operations on a closed set, add an arbitrary lower bound $w \geq w^{M I N}$ , that is, $\tilde{w} \geq {\tilde{w}}^{M I N}$ , which is equivalently $e \geq {\tilde{w}}^{M I N} - {\tilde{w}}^{M A X}$ in terms of the error. Since $e \in ℛ^{2}$ , these upper and lower bounds become the four scalar inequalities: $[\begin{matrix} 1 & 0 \\ 0 & 1 \\ - 1 & 0 \\ 0 & - 1 \end{matrix}] e \leq [\begin{matrix} 0 \\ 0 \\ w^{M A X} - w^{M I N} \\ 0 \end{matrix}],$ (22)

which define an H-representation polytope of the safe set as $S = {e | H e \leq h} .$ (23)

We then calculate a maximum positive invariant set $O_{\infty}$ , which contains all⁵⁴ the invariant sets $O \subseteq S$ such that $\tilde{w} (0) \in O \Rightarrow \tilde{w} (k) \in O \forall k \in N^{+}$ given the closed-loop dynamics of Equation (20). This iterative procedure uses the operation Pre( $A, O$ ), which generates the set of all states that evolve into a set $O : {e | P e \leq p}$ under the dynamics $A$ in one step: $P r e (A, O) = {e | P A e \leq p} .$ (24)

A set $O$ is positive invariant under $A$ if and only if $O \subseteq P r e (O)$ .⁵³ A more useful condition can be posed in terms of set equivalence: $O \subseteq P r e (O) \Leftrightarrow P r e (O) \cap O = O .$

which, with a polytope in an H-representation, can be readily checked by comparing the $P$ and $p$ of the set $O$ .

Given the safety constraint polytope $S$ in Equations (22)–(23), the well-known⁵³ Algorithm 1 returns the set of initial conditions (also expressed as a polytope) for which our system will remain within $S$ . We implemented Algorithm 1 using the MPT3 Toolbox⁵⁵ in MATLAB. Executing Algorithm 1 for our particular $A$ matrix (system identification of our actuator discussed below) with a variety of $γ < 1$ produced $O_{\infty} = S$ in every case, that is, the iteration returned in one step. This expected behavior verifies safety, confirming our intuition that as long as our supervisor's controller activates while the actuator state is $\tilde{w} \leq {\tilde{w}}^{M A X}$ , it will remain so (theoretically) for all time.

Algorithm 1. Maximum Positive Invariant Set Calculation
Input: $A$ , $S$
$O_{0} \leftarrow S$ , $O_{1} \leftarrow P r e (A, O_{0}) \cap O_{0}$ ;
while $O_{j} \neq O_{j - 1}$ do
$O_{j + 1} \leftarrow P r e (A, O_{j}) \cap O_{j}$ ;
$j \leftarrow j + 1$
end
return $O_{\infty} = O_{j}$

Supervisor integration with the nominal controller

The controller in Equation (19) drives our actuator states ( $w$ ) to a maximum safe value in closed-loop. However, we seek instead to use Equation (19) as a bound for another nominal controller, which presumably feeds back the full robot state $x$ including pose.

So, assume that there is a feedback controller developed independent of the supervisor, that would nominally close the loop of Equation (1) as $u (k) = v (x (k))$ . Propose the following composition of $v (x)$ and the supervisor's Equation (19),

u_{i}^{s} (x (k)) = m i n (v_{i} (x (k)), u_{i}^{M A X} (x (k)))

(26)

where $u_{i}^{M A X} (x (k))$ replaces the time indexing in Equation (19) with feedback, noting that the actuator state $\tilde{w}$ is part of the full state $x$ . The loop is then closed as $u (x (k)) = u^{s} (x (k))$ .

We note that this composed system is both continuous (in $C^{0}$ ) and Lipschitz continuous under certain conditions on $v_{i} (x)$ . See Section 2.2 in the Supplementary Information S1 for a full proof.

Finally, we show the most important property of $u^{s}$ , the safety verification that motivates all the work in this article.

Theorem 1. Safety. Consider the closed-loop system $x (k + 1) = f (x (k), u^{s} (x (k)))$ defined by Equations (1), (25)–(26), and the $u_{i}^{M A X}$ in Equation (19) for all actuators. If Algorithm 1 verifies that the set $S$ in Equations (22)–(23) is positively invariant for the supervisor's error dynamics of Equation (20), then $S$ is also invariant under the closed-loop dynamics of the composed system, and $({\tilde{w}}_{i} (0) - {\tilde{w}}_{i}^{M A X}) \in S \Rightarrow {\tilde{w}}_{i} (k) \leq {\tilde{w}}_{i}^{M A X} \forall k \in N^{+} .$

Proof. Invariance of $S$ under the action of the supervisor's controller alone, by assumption of the result of Algorithm 1, states that applying $u_{i}^{M A X} (x (k))$ gives $u_{i}^{M A X} (x (k)) \Rightarrow {\tilde{w}}_{i} (k + 1) \leq {\tilde{w}}_{i}^{M A X} .$ (27)

Then, by the definition of controller in Equation (26), $u_{i}^{s} (x) \leq u_{i}^{M A X} (x)$ . Let the resulting actuator state from applying $u_{i}^{s} (x (k))$ be ${\tilde{w}}_{i}^{s} (k + 1)$ . Since $h (\cdot, \cdot)$ is a monotone control system, applying $u_{i}^{s} \leq u_{i}^{M A X}$ produces $\begin{matrix} u_{i}^{s} (x (k)) \leq u_{i}^{M A X} (x) \Rightarrow {\tilde{w}}_{i}^{s} (k + 1) \leq {\tilde{w}}_{i} (k + 1) \\ \Rightarrow {\tilde{w}}_{i}^{s} (k + 1) \leq {\tilde{w}}_{i}^{M A X} . \end{matrix}$

Therefore, by induction, ${\tilde{w}}_{i} (0) \leq {\tilde{w}}_{i}^{M A X} \Rightarrow {\tilde{w}}_{i} (k) \leq {\tilde{w}}_{i}^{M A X} \forall k \in N^{+}$ .□

Hardware Testbed

The control system derived above is applicable for any soft robotic system whose equations of motion can be put in the form of Equations (1)–(3). As one particular application, this article considers feedback control of a soft robotic limb constructed with thermally actuated SMA wire coils. Versions of this limb, previously developed as part of a soft underwater robot,¹¹ have recently been deployed by the authors for both open-loop^30,56 and closed-loop⁵⁷ control as a free-standing manipulator. For eventual application in locomotion, this article uses the proposed supervisory controller to maintain safe actuator states when significant and sustained contact occurs.

Hardware design

Our soft robotic limb consists of a bulk silicone body embedded with sensors (Fig. 2) and a set of antagonistically arranged actuators. The limb is designed for planar motions only to develop algorithms with a reduced-dimensional state space. The limb's body (Smooth-On Smooth-Sil 945), shown in Figure 2a-1, has these two embedded SMA actuator coils (Dynalloy Flexinol, 0.020″ wire diameter) inserted along a horizontal ridge, as shown in Figure 2a-2, so that actuation forces cause bending deflections.

FIG. 2.

Our hardware testbed consists of a soft robot limb actuated with SMA wire coils. A cross section of the limb from Figure 1 shows (a) (1) the limb's bulk body, (2) two antagonistic SMA wire coils, (3) thermocouples attached to SMAs at the rear of the limb, and (4) a soft capacitive bending sensor. Rear angled view of the limb shows (b) (2) the SMAs constrained along a ridge on each side, (3) the thermocouples attached to the SMAs, and (4) the bending sensor in its groove at the top of the limb. A functional block diagram of the limb (c) shows the sensor measurements and control input, as measured and commanded by the controller running on a microcontroller and PC. SMA, shape memory alloy.

The two SMA wires are actuated through resistive (Joule) heating. Current through the wires was controlled using pulse-width modulation (PWM) to N-channel power MOSFET transistors connected to a 7V power supply. A microcontroller sets the PWM duty cycle between 0% and 100%, that is, each SMA's control input is $u_{i} \in [0, 1]$ .

Three sensors are located on the limb: one for the body's pose, and one each for the temperatures of the wires. Temperature is sensed by thermocouples (Omega Engineering, type K, 30 AWG) affixed to the SMA coils at the rear of the limb using thermally conductive epoxy (MG 8329TCF) via the fabrication procedure described in our prior work⁵⁶ (Fig. 2a-3, b-3). A soft capacitive bending sensor (Bendlabs, Inc.) is inserted into a groove in the limb (Fig. 2a-4, b-4) and provides a single measurement of angular deflection of the limb, $θ (t)$ , as shown in Figure 1d.

Robot and actuator model and calibration

We use a simplified model of the limb for this article, with a state space of $x = {[\begin{matrix} θ & \dot{θ} & w \end{matrix}]}^{⊤} \in ℛ^{4}, w = [\begin{matrix} T_{0} & T_{1} \end{matrix}],$ (28)

where the body pose is deflection angle, and actuator states are the temperatures of the two wires. Importantly, this article is not concerned with developing provably stabilizing controllers for the body pose, and our supervisory controller in Equation (19) does not require a model of body pose dynamics $g (\cdot)$ . Therefore, feedback of only the net deflection angle $θ$ may be sufficient for pose control as in prior work on SMA-powered robots.^32,58–60

The thermal dynamics of our SMA actuators can be approximated in the form of Equation (3). As in prior work,^30,32 the first-principles model for Joule heating in discrete time is $T_{i} (k + 1) = - \frac{h_{c} A_{c}}{C_{v}} (T_{i} (k) - T_{0}) Δ_{t} + \frac{1}{C_{v}} Δ_{t} P_{i} (k)$ (29)

for the i-th SMA at time k with specific heat capacity C_v, ambient heat convection coefficient h_c, surface area A_c, and ambient temperature T₀. The input electrical power, $P_{i} (k)$ , is current controlled with $P = ρ J^{2}$ , where $ρ$ is resistance and J is current density. For our PWM input, we assume that the duty cycle u_i modulates the fraction of time current is conducting through the SMA and approximate that current is constant when flowing, so $P_{i} (k) = ρ J^{2} u_{i} (k)$ . Substituting and factoring out unknown constants into lumped coefficients, $T_{i} (k + 1) = a_{(1, i)} T_{i} (k) + a_{(2, i)} u_{i} (k) + a_{(3, i)},$ (30)

just as in Equation (3). We calibrate Equation (30) for each SMA from data collected in hardware, using the same procedure as in our prior work on this hardware platform.^30,56

Nominal Feedback for Pose of an SMA-Actuated Soft Robot

We employ two representative nominal controllers for SMA-actuated robots to demonstrate that our supervisory control scheme is agnostic to choice of $v (x)$ . These controllers seek to modulate the robot's pose as a non-safety-critical state.

Antagonistic actuation as a SISO system

Our supervisor is derived for an arbitrary number of soft actuators (m) in the form of Equation (3). However, for our particular SMA-powered robot, prior research has shown that a pair of $m = 2$ antagonistic actuators can be reformulated as a SISO system^32,57,61 for simpler development of a $v (x)$ . We first note that the duty cycle input for each wire, $u_{i} \in [0, 1]$ , does not allow for a negative control input: we have no ability to cool the wire. However, since the actuators are oriented antagonistically, we can map one of the two SMA duty cycles in our to a negative range of a single scalar input, which we denote $v (x) = v (μ (x)),$ (31)

v (μ (x)) = \{\begin{matrix} {[μ (x) 0]}^{⊤} & i f μ (x) \geq 0 \\ {[0 - μ (x)]}^{⊤} & i f μ (x) < 0 \end{matrix}

(32)

We therefore only need to specify a (bounded) SISO nominal controller, $μ (x) : ℛ^{n} \mapsto [- 1, 1]$ .

Proportional-integral with anti-windup

We first test a proportional-integral (PI) controller. We augment it with an anti-windup (AW) block⁶² since our control $μ$ saturates at $\pm 1$ . Defining $e (k) = θ (k) - \bar{θ} (k)$ as the difference between current versus reference bending angle, implicitly indexing into $x (k)$ , and with some minor abuses of notation for clarity, our PI-AW feedback controller takes the form:

$μ (e) = s a t (η (e))$ (34)

where the linear saturation function $s a t (\cdot) : ℛ \mapsto [- 1, 1]$ is defined as $s a t (x) = \{\begin{matrix} 1 & i f x \geq 1 \\ x & i f - 1 < x < 1 \\ - 1 & i f x \leq - 1 \end{matrix}$ (35)

Therefore, $η$ is the internal state for the AW compensator, which tracks the difference between attempted versus applied control input. Tuning of the constants $K_{p}, K_{I}$ , and $K_{A W}$ is discussed in Section 3 of the Supplementary Information S1.

Sliding mode controller with boundary layer

In addition to PI feedback as a standard approach, there has been much prior success in control of SMA-based robots and mechanisms using sliding mode control (SMC).^58,59,63 SMC naturally addresses saturation issues since switching occurs between some minimum and maximum input.⁶⁴ We employ a model-free SMC with a boundary layer, as suggested by Elahinia and Ashrafiuon,⁵⁸ with a sliding surface s, using a finite-difference approximation of derivative as $\begin{matrix} ė (k) \approx \frac{1}{Δ_{t}} (e (k) - e (k - 1)), \\ s (e) : = ė (k) + 2 λ e (k) + K_{I} [\sum_{τ = 0}^{k - 1} e (τ) Δ_{t}] \end{matrix}$ (36)

μ (s) = s a t (\frac{s}{ϕ})

(37)

where $ϕ \in ℛ^{+}$ is the boundary layer thickness and $λ \in ℛ^{+}$ is the phase plane angle.⁶⁴ Tuning of this controller is also discussed in Section 3 of the Supplementary Information S1.

Supervisory Control Results

We perform three sets of tests to characterize and validate the action of the supervisor on the above controllers. To test without permanently damaging the robot, we chose temperature constraints $w^{M A X}$ of at most $9 0^{\circ} C$ , which is the transition temperature of our SMAs.

Theoretical performance verification

We first confirmed that our framework functions as intended by implementing both the PI-AW and SMC controllers in contact-free tests, and comparing their operation with versus without the supervisor. We chose an arbitrary, but aggressive, step setpoint angle ( $\bar{θ} = 4 0^{\circ}$ bending) and a maximum temperature of $w^{M A X} = 6 5^{\circ} C$ for the supervisor, with $γ = 0.2$ for conservative operation.

Figure 3 and Supplementary Video S2 show the results of all four tests. Both the PI-AW and SMC controllers, without the supervisor, regulate the limb around the desired setpoint with low error. However, both controllers cause the SMA wire temperatures to drift upwards, representing potentially unsafe operation. In contrast, the controllers with the supervisor cause temperature to saturate at the maximum. This is the intended behavior: the supervisor's activation sacrifices state tracking in favor of safe actuator states and $w \leq w^{M A X}$ for all actuators. We conclude from these tests that the supervisor indeed operates independent of the underlying nominal controller $v (x)$ .

FIG. 3.

The two different nominal controllers (PI-AW in red, SMC in orange) stabilize around the desired angle but may overheat the limb. However, imposing the supervisor ensures safe operating temperatures (blue, green) while attempting to reach the control goal. PI-AW, proportional-integral-anti-windup; SMC, sliding mode control.

Supervisory controller tuning

Our next test, with $\bar{θ} = 3 0^{\circ}$ and a maximum temperature of $w^{M A X} = 6 0^{\circ} C$ , executed the PI-AW controller with the supervisor for various values of its parameter $γ$ . We only test using the PI-AW nominal controller for pose since Figure 3 shows similar activation behaviors for both nominal controllers.

The data in Figure 4 and Supplementary Video S3 demonstrate large variations in behavior depending on $γ$ . For small values ( $γ \in [0.05, 0.2]$ ), the supervisor activates almost as soon as control begins ( $t = 5$ s), and temperature trajectories rise slowly but remain safe. For larger values ( $γ \in [0.3, 0.9]$ ), the supervisor is much less aggressive. For the largest values, the limb briefly reaches the target $\bar{θ}$ before the supervisor forces a lower input. We do not report a $γ = 1$ result since its performance was poor.

FIG. 4.

Controller tuning shows that large values of the parameter $γ \in (0, 1)$ , when the actuator dynamics are not known exactly, may slightly violate safety constraints. For practical uses, we found that $γ < 0.3$ demonstrated safe operation.

At larger values of $γ$ , some violation of the safety constraint is observed, which is expected given our system identification of $A$ in Equation (30). Unmodeled dynamics cause the value for $u^{M A X}$ to be artificially large. This suggests that smaller $γ$ values are best in practice since soft robot modeling is often imprecise.

Physical interactions

We finally stress-test our feedback method in three different physical interaction scenarios, each representing an eventual use of our soft limb. These three scenarios, in Figure 5, include environmental contact, human contact, and the attempted tracking of infeasible/unsafe trajectories. All tests again used the PI-AW nominal controller with $γ = 0.2$ .

FIG. 5.

Three physical interactions that could cause damage to a soft robot under feedback: (a) contact and collision with kinematic constraints, such as a wall or floor, (b) unmodeled disturbances such as human interaction, and (c) attempting infeasible motions such as might unintentionally occur in learning from demonstration tasks.

The first test (Fig. 5a) places a wall next to the limb that blocks it from reaching its target bend angle. In the second test (Fig. 5b), a human pushes on the robot causing a disturbance. The third test (Fig. 5c) tracks a trajectory of bending angles recorded beforehand by a human operator moving the limb, as in our prior work.³⁰ Substituting the recorded trajectory as ${\bar{θ}}_{1, \dots, K}$ attempts to recreate that motion, however unsafe it may be. With the supervisor, this procedure can be viewed as a crude form of learning from demonstration,⁶⁵ where a feedback controller mimics a demonstrated action under safety or feasibility constraints.

The data from these tests are in Figure 6, and demonstrations are shown in Supplementary Videos S4–S6. For the wall interaction, the unsafe control system without the supervisor heats the SMA wire rapidly and was manually deactivated before the test concluded, whereas the supervisor keeps the actuator at a steady maximum temperature. For the human disturbance, the unsafe controller responded dynamically to the disturbance, causing continued heating, whereas the test with the supervisor prevented a changing input during those motions and implicitly bounded the force applied to the human. Finally, for the “learning from demonstration” test, the unsafe controller was able to faithfully track the desired motion; however, the SMA temperatures violated constraints. The corresponding test with the supervisor demonstrates it dynamically activating and deactivating as both wires reach potentially unsafe operation.

FIG. 6.

In each of the three physical interaction examples, the supervisory controller ensures safe operation when the robot's actuator state would otherwise violate safety constraints. PWM, pulse-width modulation.

Discussion

This article proposes a supervisory control scheme for a generalizable class of soft robot actuators, provably verifying that actuator states remain in a safe region. The proposed supervisor is simple to formulate and implement, with very low online computational cost. Experiments show that the controller can be tuned for conservative operation even in the case when the actuator dynamics are a significant approximation, making the framework applicable for a variety of soft robot actuator designs and modalities. We demonstrate that the controller safely operates on a thermal actuator in hardware tests, maintaining safe temperatures in a variety of contact-rich environments.

This work highlights the inherent relationship between force applied at a manipulator tip and the bounds on its actuator state, for example, in the human contact disturbance test. Recent work has shown that environmental contact forces for a soft robot may be estimated simply from pose measurements.⁶⁶ Therefore, if a model for the body dynamics is available, it may be possible to convert between actuator bounds and body force bounds, allowing the concepts from this article to extend to safe interactions of body-to-environment: safety in pose as well as safety in actuator.

Future Work and Conclusions

Multiple directions of future work are anticipated to make the proposed framework more robust and applicable with fewer assumptions required. In particular, a probabilistic actuator dynamics model, and accompanying modifications to the controller, may provide better robustness when the linearization is poor. Similarly, future work will examine adaptive control for capturing unmodeled dynamics. If the actuator dynamics cannot be linearized, we will examine nonlinear optimization techniques for the supervisor, such as model-predictive control. For robots with coupled actuator dynamics or more than one scalar state per actuator, future work may saturate actuators in terms of conic section bounds.⁵¹ And although many soft robot actuators are monotonic control systems, the soft robot itself may not be, prompting future work in extending, for example, optimization-based approaches to safety in nonmonotone systems.⁶⁷

The system in this article relies on feedback for the supervisor, requiring sensors for all actuator states. However, if the actuator dynamics model sufficiently captures the underlying physical phenomena, it may be possible to estimate the states $\tilde{w} (k)$ for use in the supervisor, eliminating the need for external sensing.

Finally, a major motivation of this article is applying feedback control to soft robots in locomotion and human–robot interaction tasks. We plan to implement our supervisor on SMA-actuated walking soft robots¹¹ to demonstrate safe, closed-loop locomotion in state-feedback. Safe locomotion with feedback will bring soft robots closer to real-world deployment and increase the acceptance of soft robots for real-world tasks.

Footnotes

Acknowledgments

We thank Xiaonan Huang, Richard Desatnik, and all members of the Soft Machines Laboratory at Carnegie Mellon University for their collaboration in the design framework for the robot studied in this article.

Authors' Contributions

A.P.S.: Conceptualization, formal analysis, methodology, software, funding acquisition, writing—original draft, review and editing. Z.J.P.: Methodology, software, writing—review and editing. A.T.W.: Methodology, software, writing—review and editing. C.M.: Funding acquisition, supervision, writing—review and editing.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was in part supported by the Office of Naval Research under Grant No. N000141712063 (PM: Dr. Tom McKenna), the National Oceanographic Partnership Program (NOPP) under Grant No. N000141812843 (PM: Dr. Reginald Beach), and an Intelligence Community Postdoctoral Research Fellowship through the Oak Ridge Institute for Science and Education.

Supplementary Material

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Supplementary Material

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Safe Supervisory Control of Soft Robot Actuators

Abstract

Introduction

Background: robot safety

Soft actuator safety and degradation

Approach and applicability

Supervisory Control for a Soft Robotic Actuator

System model

Static bounds on control input are impractical for safety

The supervisor's dynamically saturating controller

Safety verification of the supervisor's controller

Supervisor integration with the nominal controller

Hardware Testbed

Hardware design

Robot and actuator model and calibration

Nominal Feedback for Pose of an SMA-Actuated Soft Robot

Antagonistic actuation as a SISO system

Proportional-integral with anti-windup

Sliding mode controller with boundary layer

Supervisory Control Results

Theoretical performance verification

Supervisory controller tuning

Physical interactions

Discussion

Future Work and Conclusions

Footnotes

Acknowledgments

Authors' Contributions

Author Disclosure Statement

Funding Information

Supplementary Material

References

Supplementary Material