Loop Shaping of Hybrid Motion Control with Contact Transition

Motion control systems that can come in a well specified or (more importantly) unforseen contact with environmental objects have always been the focus of active research, especially in the field of control and robotics, starting already in the eighties of the last century. For instance, dynamics stability issues during contact with stiff environments were recognized and addressed (often in context of industrial robotics), see [1], when a manipulator in the force control mode comes in touch with a stiff and kinematically constrained environmental object. A milestone was the introduction of the concept of impedance control [2] and later of hybrid impedance control [3], which enabled a deep understanding of the most important (i.e., physically justifiable) interactions and limitations for the controlled impedance–admittance pair of a mechanical motion system in contact with its environment. Motion control of an unconstrained manipulation, on the one hand, and force control of a constrained interaction between the manipulator and its environment, on the other hand, became quickly to ’stumbling block’, especially in view of controlling contact transitions, see e.g. discission with experiments in [4]. For a former brief survey of the force control of manipulators we refer to e.g. [5], while a more recent overview of the force control can be found in the literature like e.g. Springer Handbook of Robotics [6].

The ideas of impedance and admittance control of robotic manipulators, [2, 3], found quickly a way and appreciation in motion control for drives and mechatronic systems [7]. A unified passivity-based control framework for position, torque and impedance control, which use the full state feedback and a dedicated energy shaping with variable gain strategy was also proposed in [8]. Another focus on decomposing (correspondingly switching) the control structure led to hybrid position/force controllers, the first one proposed (most probably) in [9]. A widely adopted strategy of hybrid force/motion control, which aims at controlling the motion along the unconstrained task directions and force along the constrained task directions, is using a certain decomposition which allows simultaneous control of both the contact force and the end-effector motion in two mutually independent subspaces, see [6] for details. Here, a selection strategy for the stiffness/compliance parameters and the desired and feedback variables must be part of the overall control law and is known to be non-trivial and fundamental to the specification of the control task. Although impedance modulation and reconfiguration (corresponding switching) are widely used in hybrid position/force control in robotics, see e.g. [10], equally as in other motion control systems such as hydraulic actuators [11], the problems of transition and stability of structural switching [12] remain among the most relevant. It should be emphasized that during a contact transition, both a hybrid position/force control and the process plant itself undergo a structural change. For sufficiently damped contact transitions and relatively slow dynamics of some available internal state variable experiencing a threshold value upon the contact, a hysteresis relay-based switching policy can be developed, see e.g. [13]. For combining the robustness property of impedance control in stiff contact with the accuracy of admittance control in soft contact various approaches were proposed in robotics. For instance, a predictive instantaneous model impedance control scheme was described in [14], and continuous switching (with duty cycle as design parameter) between controllers with impedance and admittance causality was provided, also with experiments, in [15].

An important causality constraint is that no one motion system can simultaneously impress a force on its environment and impose a displacement or velocity on it. Only one of both control variables, either interaction force or relative motion of the environmental object, can be determined along each degree of freedom [2]. This results from an instantaneous power flow between two or more physical systems. Recall that the power flow is definable as product of an effort and a flow variable, force and velocity for mechanical systems, respectively. Given these basic principles, it seems obvious that different control concepts and an approach for combining them are required to control unconstrained and constrained motion and the resulting contact force.

A frequently appearing question is nevertheless how to handle those manipulator-environment configurations where a control design is a priori constrained by some given structure, the application specification and, most importantly, by the sensors and their arrangement.

In this work, an intuitively understandable (for standard motion control design in frequency domain) approach of reshaping the otherwise stiffly designed feedback controllers is proposed. This way, a smooth and stable contact transition can be guaranteed without applying a more complex control structure. Most importantly, only the measured output displacement is used for feedback and the designed hybrid motion control does not require additional force sensors or observers of internal states. The rest of the paper is organized as follows. In section II we discuss the types of impedance operators that represent an interaction with the environment and relate them to the control loop properties required for a contact transition. Section III introduces the thereupon based hybrid motion control design. A detailed experimental case study with soft but penetrable grape objects coming into contact with the mechanical tool which is driven by the displacement feedback control is demonstrated in section IV. Brief conclusions are drawn in section V.

For motion control at large, with one relative degree of freedom $x$ in the generalized coordinates, one can define the control stiffness, cf. [7], [16], as

$$\kappa=\frac{\partial f(\cdot)}{\partial x(t)}\biggl{|}_{t\rightarrow\infty}.$$

(1)

Here $f(\cdot)$ is the overall net force (in the generalized coordinates) imposed on the moving body of the motion control system under consideration. For a matched disturbance force $F(t)$ and a feedback control value $u(t)$, which inherently has dimension of a force in motion control, the total net force can be considered as a superposition $f=u+F$. Indeed, when a generic motion control system

$$\dot{y}=g(y,u,t),$$

(2)

with the sufficiently smooth flow map $g(\cdot)$ and the vector of states variables $y\in\mathbb{R}^{n}$ (often with $n=2$ and $y=(x(t),\dot{x}(t))^{\top}$ as a vector of relative displacement and velocity) is in steady-state, the control should balance the counteracting disturbance. Let us fix next some controlled operation point, in terms of $(u_{0},x_{0})$, and consider this way reduced (1) in Laplace domain. Then, one can write

$$\tilde{\kappa}(s)=\frac{F(s)}{x(s)},$$

(3)

provided that the Laplace transform of the control stiffness operator exists. Also we note that an equivalence between $\kappa$ and $\tilde{\kappa}$ is valid only for low frequency range, cf. [3]. Recalling that for a linear environment, the impedance is defined as the ratio of the Laplace transforms of effort and flow [17], the mechanical impedance is written as:

$$Z(s)=\frac{\hbox{\emph{force}}(s)}{\hbox{\emph{velocity}}(s)},$$

(4)

that leads to a generic relationship

$$\tilde{\kappa}(s)=Z(s)s.$$

(5)

Since a dynamic interaction between two physical bodies implies that one must complement the other, i.e. along any degree for freedom if one is an impedance, the other must be admittance and vice versa, see [2], the condition (5) becomes crucial. It reveals how the motion control system can be designed when an environment is classified by $Z(s)$.

Using the mechanical impedance definition (4), one can classify the following (typical) contact environments, which are associated with the corresponding impedance operators. The first one is the viscous dashpot, shown schematically in Fig. LABEL:fig:environm (a).

The constitutive equation of the Newtonian fluid in a dashpot, i.e. $F=\alpha\,\dot{x}$ where $\alpha>0$ is viscosity, results in $Z_{v}(s)=\alpha$, meaning the contact environment is resistive, cf. [17]. If one expects the environment to be viscoelastic, i.e. to exhibit also a certain capacitive behavior, then one can assume a Kelvin-Voigt contact, that is schematically shown in Fig. LABEL:fig:environm (b). This leads to the corresponding impedance operator $Z_{ve}(s)=\alpha+\beta/s$. Note that for modeling the environmental impedance, more complex structures than those shown in Fig. LABEL:fig:environm can equally be assumed, and even variable and nonlinear structures might be considered. This would, however, go far beyond the scope of this work, and it turns out to be superfluous for a straightforward design of linear and hybrid motion controllers.

Now, consider the loop transfer function

$$L(s)=C(s)G(s)$$

(6)

of the motion control system with the input-output plant $G(s)=x(s)(u(s)+F(s))^{-1}$ and feedback controller $C(s)$ which receives the control error $e(s)=r(s)-x(s)$ as input. Obviously, the control reference command $r(s)$ conforms to some application-related specifications, before and after a possible contact with environment and without it. While the reference-to-output transfer function $H(s)=L(s)(1+L(s))^{-1}$ can be determined by shaping $L(s)$, correspondingly designing $C(s)$ to be possibly stiff, i.e. having possibly high bandwidth of $H(s)$ and possibly unity $|H(j\omega)|$ for all angular frequencies $\omega\in(0,+\infty)$, we focus in the following on the disturbance response to $F(s)$. The disturbance-to-output transfer characteristics are given by

$$S(s)=\frac{x(s)}{F(s)}=\frac{G(s)}{1+L(s)}$$

(7)

and often denoted and disturbance sensitivity function, cf. e.g. [18]. While an ideal position (or velocity) controller should not allow any steady-state or transient deviations for any force imposition on mechanical system, i.e. the controller stiffness should be infinite, cf. [7], a hybrid motion controller with contact transition should allow $S(s)$ to match the $Z(s)$ properties of the contact with environment. Comparing (3), (5), and (7) one can recognize that

$$S(s)=\tilde{\kappa}^{-1}(s)=\frac{1}{Z(s)s}.$$

(8)

To further interpret the results obtained above, it is worth recalling a fundamental distinction between a mechanical admittance and impedance [2]. Multiple physical systems can be described in one form but not in the other. For instance, elastic contacts approximated by a non-monotonic constitutive equation can only be seen as impedance, i.e. $x\mapsto F$, but not as admittance. Indeed, prior to a mechanical contact is established, the map $F\mapsto x$ is not given. Similar issue appears in case of a tangential kinetic friction force, cf. [19]. Nevertheless, the admittance of the motion control, which is equivalent to disturbance sensitivity function (7), can certainly be used at the same moment as the mechanical contact with environment arises. This will be discussed and used further below in the derivation of hybrid motion control.

First, we proceed with designing a (sufficiently) stiff feedback motion control, denoted by $C_{s}$. Following the most simple loop shaping methodology and assuming the critically damped dominant pole pair of the closed-loop system, the reference-to-output transfer function yields

$$H(s)=\frac{L(s)}{1+L(s)}=\frac{\omega_{0}^{2}}{s^{2}+2\omega_{0}s+\omega_{0}^{2}}.$$

(9)

Here the control specification is given by only the natural frequency $\omega_{0}$ (approximately to bandwidth) of the closed-loop. Recall that higher $\omega_{0}$ values imply higher stiffness of the controlled system. Using the given system transfer function $G(s)$ and applying the block diagram algebra, the resulting motion control $C_{s}(s)=L(s)G^{-1}(s)$ is

$$C_{s}(s)=\frac{\omega_{0}^{2}}{G(s)s(s+2\omega_{0})}.$$

(10)

Note that for system plants $G(s)$ with a relative degree $\leq 2$, the control (10) yields a proper transfer function and, thus, can be directly implemented.

Now, for designing an impedance controller, assume a sensitivity function, cf. (8),

$$S_{v}(s)=\frac{1}{\alpha s},$$

(11)

that corresponds to a viscous (dashpot-type) contact with environment, cf. section II. Using (7) and the block diagram algebra, the resulting impedance controller, denoted as viscous $C_{v}(s)$, is determined by

$$C_{v}(s)=\frac{\alpha s\,G(s)-1}{G}.$$

(12)

Note that (12) reveals an improper transfer function so that an additional low-pass filter with a sufficiently high cut-off frequency needs to applied in series with $C_{v}(s)$.

The magnitude response of $G(j\omega)$, $H(j\omega)$, $S_{s}(j\omega)$, and $S_{v}(j\omega)$ transfer functions are depicted in Fig. 2 for the exemplary assumed system plant with two negative real poles at $\{-1000,-10\}$, and the design parameters $\omega_{0}=100$ and $\alpha=100$. Note that both disturbance sensitivity functions are determined according to (7) for each of the feedback controllers $C_{s}$ and $C_{v}$.

It is easy to interpret from the Bode plot of $S_{s}(j\omega)$ that a step-wise disturbance $F(s)$, which appears at the contact instant, will be largely suppressed by the stiff motion control. Following to that, the controlled mechanical motion system will penetrate into the environmental object or moves it away from itself if the latter is unconstrained. Quite the opposite, the viscous impedance control will react to the step-wise contact force $F$ by inducing a repulsive relative displacement in the opposite direction. Here it is worth noting that since it leads to release of the contact, the disturbance becomes zero and the relative motion will stop, cf. below with the experimental results.

Recall that for achieving stable and smooth contact transitions, a general strategy of the motion control is to regulate the system displacement and/or velocity (as conventional manipulators do) and provide additionally a well-specified disturbance response for deviations from this motion. According to [2], such disturbance response has the form of an impedance, that may be then modulated and adapted depending on the control tasks and environment. Despite such straightforward impedance paradigm, that gives the name ’impedance control’ [2], one task that is not always solvable remains the detection of contact, correspondingly recognition of the associated deviations from a well-specified (i.e. nominal) motion. Mostly, the contact forces are measured by a force sensor connected to the wrist of manipulator or integrated in the front-end tool. The use of internal (i.e. not only output displacement) measurements or even external senors can be found both in the theoretical works, cf. e.g. [20] and experimental studies in robotics, e.g. [21, 10].

If the controlled output displacement is the only measurable state of the motion system (the case we consider in this work), the control reshaping can be triggered exclusively by an information content of the control signal. Assuming some nominal bound of the control signal $U$, that is mostly possible for the given nominal plant $G(s)$, control $C_{s}(s)$, and reference $r(s)$, an overshot $|u(t)|>U$ will indicate appearance of the disturbance force $F$. Note that this strategy can be used in particular when the stiff motion controller $C_{s}$ contains an integral control action. Indeed, during a compensated steady-state motion, an exceeded control force is proportional to an additional disturbing force. Since a dynamic transition from $C_{s}$ to $C_{v}$ should not provoke any undesired transients in direction of the contact with environment, it is worth examining the disturbance-to-control-value transfer characteristics which are given by $U(s)=u(s)/F(s)=C(s)G(s)(1+C(s)G(s))^{-1}$. For both controllers (10) and (12), as exemplary designed above, this is shown by the magnitude response in Fig. 3.

One can recognize that since $r$ is set to zero for $C_{v}$, and the control magnitude response of both $C_{s}$ and $C_{v}$ have nearly the same value, an appearance of the step-wise disturbance $F(t_{c})$ at $t=t_{c}$ will lead to a decrease of $|u(t_{c})|=U$ by the magnitude equal to $|F|$ for $t>t_{c}$. The force imposed on the environmental object under contact drops respectively.

In this communication, we provided an easy interpretable (in frequency domain) analysis and design of dynamic reshaping of the stiff (admittance) motion control to a soft (impedance) control upon the contact with constrained and deformable environmental objects. Only the measured output displacement in feedback is used for the overall control design and operation. Hybrid control scheme was established and revealing control experiments with contacting the soft but penetrable objects, i.e. grape, were provided.