A family of virtual contraction based controllers for tracking of flexible-joints port-Hamiltonian robots: theory and experiments

02/04/2020 ∙ by Rodolfo Reyes-Báez, et al. ∙ IEEE 0

In this work we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots (FJRs) in the port-Hamiltonian (pH) framework. The proposed design method, called virtual contraction based control (v-CBC), combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity and the existence of (incrementally) passive maps.

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1 Introduction

Control problems in rigid robots have been widely studied in the literature due to they are instrumental in modern manufacturing systems. However, as pointed out in Tomei 1

the elasticity in the joints often can not be neglected for accurate position tracking. For every joint that is actuated by a motor, we basically need two degrees of freedom instead of one. Such FJRs are therefore

underactuated mechanical systems. In the work of Spong2 two state feedback control laws based, respectively, on feedback linearization and singular perturbation theory are presented for a simplified FJRs model. Similarly, in Canudas3 a dynamic feedback controller for a more detailed model is presented. In Loría 4 a computed-torque controller for FJRs is designed, which does not need jerk measurements. In Ortega5 and Brogliato6 passivity-based control (PBC) schemes are proposed. The first one is an observer-based controller which requires only motor position measurements. In the latter one, a PBC controller is designed and compared with backstepping and decoupling techniques. For further details on PBC of FJRs we refer to Ortega et al. 7 and references therein. In Astolfi 8, a global tracking controller based on the immersion and invariance (I&I) method is introduced.
From a practical point of view, in Albu-Schäffer 9, a torque feedback is embedded into the passivity-based control approach, leading to a full state feedback controller, where acceleration and jerk measurements are not required. In the recent work of Ávila-Becerril 10, a dynamic controller is designed which solves the global position tracking problem of FJRs based only on measurements of link and joint positions. In the work of 11 an adaptive-filtered backstepping design is experimentally evaluated in a single flexible-joint prototype. All of these control methods are designed for FJRs modeled as second order Euler-Lagrange (EL) systems. Most of these schemes are based on the selection of a suitable storage function that together with the dissipativity of the closed-loop system, ensures the convergence of the state trajectories to the desired solution.

As an alternative to the EL formalism, the pH framework has been introduced in van der Schaft12. The main characteristics of the pH framework are the existence of a Dirac structure (connects geometry with analysis), port-based network modeling and the clear physical energy interpretation. For the latter part, the energy function can directly be used to show the dissipativity of the systems. Some set-point controllers have been proposed for FJRs modeled as pH systems. For instance in Borja 13 the controller for FJRs modeled as EL systems in Ortega7 is adapted and interpreted in terms of the Control by Interconnection technique111We refer interested readers on CbI to 14. (CbI). In Zhang15, they propose an Interconnection and Damping Assignment PBC (IDA-PBC222For IDA-PBC technique see also 16.) scheme, where the controller is designed with respect to the pH representation of the EL-model in Albu-Schäffer9.

For the tracking control case of FJRs in the pH framework, to the best of our knowledge, the only results available in the literature are the singular-perturbation approach in Jardón-Kojakhmetov18 and our preliminary work Reyes-Báez19.

In the present work we propose a setting that extends our previous results in Reyes-Báez20 and Reyes-Báez21 on v-CBC of fully-actuated mechanical systems to solve the tracking problem of FJRs modeled as pH systems. This method relies on the contraction properties of the so-called virtual system, see the works22, 23, 24, 25, 26. Roughly speaking, the method333The use of virtual systems for control design was already considered in 27 and 28. consists in designing a control law for a virtual system associated to the original FJR, such that the closed-loop virtual system is contractive and a predefined reference trajectory is exponentially stable. Finally, this control scheme is applied to the original FJR. It follows that the reference trajectory of the virtual system and the original state converge to each other.

The paper is organized as follows: In Section 2, the theoretical preliminaries on virtual contraction based control (v-CBC) and key properties of mechanical systems in the pH framework are presented. Section 3 presents the pH model of FJRs, together with the statement of the trajectory tracking problem and its solution. The main result on the construction of a family v-CBC schemes for FJRs are presented in Section 4. In Section 5, the performance of two v-CBC tracking controller is evaluated experimentally on a two-degrees of freedom FJR. Finally, in Section 6 conclusions and future research are stated.

2 Preliminaries

2.1 Contraction analysis and differential passivity

In this section, the differential approach to incremental stability29 by means of contraction analysis is summarized. Sufficient conditions in terms of the frameworks of the differential Lyapunov theory 22 and of the matrix measure25 are given. These ideas are later extended to systems having inputs and outputs with the notion of differential passivity30, and to virtual control systems26, 27. For a self-contained and detailed introduction to these topics see also 31.

Let be an -dimensional state space manifold with local coordinates and tangent bundle . Let and be the input and output spaces, respectively. Consider the nonlinear control system , affine in the input , given by

(1)

where , and

. The time varying vector fields

, for and the output function are assumed to be smooth. System in closed-loop with the state feedback defines the system given by

(2)

Solutions to system are given by the trajectory from the initial condition , for a fixed initial , at time , with . Consider a simply connected neighborhood of such that is forward complete for every , i.e., for each , each and each . Solutions to are defined in a similar manner and are denoted by . By connectedness of , any two points in can be connected by a regular smooth curve , with . A function is said to be of class if it is strictly increasing and 32. When it is clear from the context, some function arguments will be left out in the rest of this paper.

Definition 1 (Incremental stability 22).

Let be a forward invariant set, be a continuous metric and consider system given by (2). Then, system is said to be

  • Incrementally stable (-S) on (with respect to ) if there exist a function such that for each , for each and for all ,

    (3)
  • Incrementally asymptotically stable (-AS) on if it is -S and for all , and for each ,

    (4)
  • Incrementally exponentially stable (-ES) on if there exist a distance , , and such that for each , fir each and for all ,

    (5)

Above definitions are the incremental versions of the classical notions of stability, asymptotic stability and exponential stability 32. If , then we say global -S, -AS and -ES, respectively. All properties are assumed to be uniform in .

2.1.1 Differential Lyapunov theory and contraction analysis

Definition 2.

The prolonged33 control system associated to the control system in (1) is given by

(6)

with , , and . The prolonged system of in (2) is similarly defined as

(7)
Definition 3.

A function is a candidate differential or Finsler-Lyapunov function if it satisfies

(8)

for some , and with a positive integer where is a Finsler structure22, uniformly in and .

The relation between a candidate differential Lyapunov function and the Finsler structure in (8) is a key property for incremental stability analysis, since it implies the existence of a well-defined distance on via integration as defined below.

Definition 4.

Consider a candidate differential Lyapunov function on and the associated Finsler structure . For any subset and any , let be the collection of piecewise curves connecting and with and . The Finsler distance induced by the structure is defined by

(9)

The following result gives a sufficient condition for incremental stability in terms of differential Lyapunov functions. [Direct differential Lyapunov method 22] Consider the prolonged system in (7), a connected and forward invariant set , and a function . Let be a candidate differential Lyapunov function satisfying

(10)

for each uniformly in . Then, system in (2) is

  • incrementally stable on if for each ;

  • Incrementally asymptotically stable on if is a function;

  • incrementally exponentially stable on if .

Definition 5.

We say that contracts22 (respectively does not expand34 ) in if (10) is satisfied for a function of class (resp. for all ). The set is the contraction region (resp. nonexpanding region).

[Riemannian contraction metrics] The so-called generalized contraction analysis in Lohmiller24 with Riemannian metrics can be seen as a particular case of Theorem 4 as follows: Take as candidate differential Lyapunov function to

(11)

where , and is smooth and positive for all . If

(12)

holds for all , uniformly in , then, contracts (11). Condition (12) is equivalent to verify that the generalized Jacobian24

(13)

satisfies22, 35 uniformly in , where is a matrix measure444Given a vector norm on a linear space, with its induced matrix norm , the associated matrix measure is defined25 as the directional derivative of the matrix norm in the direction of

and evaluated at the identity matrix, that is:

, where is the identity matrix. as shown by Russo 36, Forni 22 and Coogan 35.

2.1.2 Differential passivity

Definition 6 (van der Schaft30, Forni37).

Consider a nonlinear control system in (1) together with its prolonged system given by (6). Then, is called differentially passive if the prolonged system is dissipative with respect to the supply rate , i.e., if there exist a differential storage function function satisfying

(14)

for all uniformly in . Furthermore, system (1) is called differentially lossless if (14) holds with equality.

If additionally, the differential storage function is required to be a differential Lyapunov function, then differential passivity implies contraction when the variational input is . For further details we refer to the works of van der Schaft30 and Forni 38.

The following lemma characterizes the structure of a class of control systems which are differentially passive. [Reyes-Báez21] Consider the control system in (1) together with its prolonged system in (6). Suppose there exists a transformation such that the variational dynamics in (6) given by

(15)

takes the form

(16)

where

is a Riemannian metric tensor,

, are rectangular matrices. If condition

(17)

holds for all uniformly in , with of class . Then, is differentially passive from to with respect to the differential storage function given by

(18)

The passivity theorem of negative feedback interconnection of two passive systems resulting in a passive closed-loop system can be extended to differential passivity as follows. Consider two differentially passive nonlinear systems , with states , inputs , outputs and differential storage functions , for . The standard feedback interconnection is

(19)

where denote external outputs. The equations (19) imply that the variational quantities satisfy

(20)

The variational feedback interconnection (20) implies that the equality holds. Thus, the closed-loop system arising from the feedback interconnection in (20) of and is a differentially passive system with supply rate and storage function , as it is shown by van der Schaft 30.

2.1.3 Contraction and differential passivity of virtual systems

Definition 7 (Reyes-Baez21, Wang24).

Consider systems and , given by (1) and (2), respectively. Suppose that and are connected and forward invariant. A virtual control system associated to is defined as

(21)

with state and parametrized by , where and are such that

(22)

Similarly, a virtual system associated to is defined as

(23)

with state and parametrized by , where and satisfying

(24)

It follows that any solution of the actual control system in (1), starting at for a certain input , generates the solution to the virtual system in (21), starting at with , for all . In a similar manner for the closed actual system in (2), any solution starting at , generates the solution to the closed virtual system in (23), starting at , for all . However, not every virtual system’s solution corresponds to an actual system’s solution. Thus, for any trajectory , we may consider (21) (respectively (23)) as a time-varying system with state .

[Virtual contraction 26, 22] Consider systems and given by (2) and (23), respectively. Let and be two connected and forward invariant sets. Suppose that is uniformly contracting with respect to . Then, for any initial conditions and , each solution to converges asymptotically to the solution of . If the conditions of Theorem 7 hold, then system is said to be virtually contracting. If the virtual system is differentially passive, then the system is said to be virtually differentially passive. In this case, the steady-state solution is driven by the input and is denoted by . This last property can be used for v-CBC, as will be shown later.

2.1.4 Virtual contraction based control (v-CBC)

From a control design point of view, the usual task is to render a specific solution of the system exponentially/asymptotically stable, rather than the stronger contractive behavior of all system’s solutions. In this regard, as an alternative to the existing control techniques in the literature, we propose a design method based on the concept of virtual contraction to solve the set-point regulation or trajectory tracking problems. Thus, the control objective is to design a scheme such that a well-defined Finsler distance between the solution starting at and desired solution shrinks by means of virtual system’s contracting behavior.

The proposed design methodology is divided in three main steps:

  1. Propose a virtual system (21) for system (1).

  2. Design a state feedback for the virtual system (21), such that the closed-loop system is contractive and tracks a predefined reference solution.

  3. Define the controller for the actual system (1) as .

If we are able to design a controller with the above steps, then, according to Theorem 7, all the solutions of the closed-loop virtual system will converge to the closed-loop original system solution starting at , that is, as .

2.2 A class of virtual control systems for mechanical systems in the port-Hamiltonian framework

In this subsection, the previous notions on contraction and differential passivity are applied to mechanical systems described in the port-Hamiltonian framework12.

2.2.1 Port-Hamiltonian formulation of mechanical systems

A port-Hamiltonian system with dimensional state space manifold , input and output spaces , and Hamiltonian function , is given by

(25)

where is a matrix, is the interconnection matrix and is the positive semi-definite dissipation matrix.

In the specific case of a mechanical system with generalized coordinates on the configuration space of dimension and velocity , the Hamiltonian function is given by the total energy

(26)

where is the state, is the potential energy, is the momentum and the inertia matrix is symmetric and positive definitive. Then, the pH system (25) takes the form

(27)

with matrices

(28)

where is the damping matrix and and are the identity, respectively, zero matrices. The input force matrix has rank ; if we say that the mechanical system is underactuated, otherwise it is fully-actuated. System (27) defines the passive map with respect to the Hamiltonian (26) as storage function.

Using the structure of the internal workless forces, system (27) can be equivalently rewritten as, see Reyes-Báez19, 31.

(29)

where , and

is a skew-symmetric matrix whose

-th element is555The structure of matrix is a consequence of the fact that Hamilton’s principle is satisfied. This was first reported by Arimoto and Miyazaki 39.

(30)

From the energy balance along the trajectories of (29), it is easy to see that forces are workless, i.e., their power is zero. Thus, system (29) preserves the passivity property of the map , as well with (26) as storage function.

2.2.2 A class of virtual control systems for mechanical pH systems

Let be the state of system (27). Following Definition 7 and considering the port-Hamiltonian formulation (29) of (27), we construct the virtual mechanical control system associated to (27) as the time-varying system given by19

(31)

with state , parametrized by the state trajectory of (29), and with Hamiltonian-like function

(32)

where . Remarkably, the virtual control system (31) is also passive with input-output pair and -parametrized storage function (32), for every state trajectory of (29). Furthermore, system (31) can be rewritten as

(33)

with as in (28) and matrices

(34)

where and . The skew-symmetric matrix defines an almost-Poisson tensor31 implying that energy conservation is satisfied. However, system (33) is not a pH system since does not necessarily hold. Thus, we refer to system (33) as a mechanical pH-like system. The variational virtual dynamics of system (33) is

(35)

Notice that (35) is of the form (16) with , and . Moreover, if hypotheses in Lemma 6 are satisfied, then system (31) is differentially passive with supply rate .

3 Problem Statement

3.1 Flexible-joints robots as port-Hamiltonian systems

FJRs are a class of robot manipulators in which each joint is given by a link interconnected to a motor through a spring; see Figure 1. Two generalized coordinates are needed to describe the configuration of a single flexible-joint, these are given by the link and motor positions as shown in Figure 1.

Figure 1: Flexible joint mechanical structure: motor’s shaft position , spring’s deflection and link’s position .

Thus, FJRs are a class of underactuated mechanical systems of degrees of freedom (dof). The dof corresponding to the -motors position are actuated, while the dof corresponding to the links position are underactuated, with . We consider the following standard modeling assumptions in Spong 2 and Jardón-Kojakhmetov18:

  • The deflection/elongation of each spring is small enough so that it is represented by a linear model.

  • The -th motor driving the -link is mounted at the -link.

  • Each motor’s center of mass is located along the rotation axes.

The FJR’s generalized position is split as , the inertia and damping matrices are assumed to be block partitioned as follows

(36)

where and are the link and motors inertia matrices, and and are the link and motor damping matrices. The total potential energy is given by

(37)

with links potential energy , motors potential energy and the (coupling) potential energy due to the joints stiffness . The corresponding potential energy for linear springs is

(38)

with and the stiffness coefficients matrix is symmetric and positive definitive. Since , the input matrix is given as . Substitution of the above specifications in the Hamiltonian function (26) and the pH mechanical system (28) results in the port-Hamiltonian model for a FJR explicitly given by

(39)

where and are the links and motors momenta, respectively; and . Without loss of generality we take . The pH-FJR (39) can be rewritten as the alternative model (29) with

(40)

with and . We will also denote the state of (39) by .

3.2 Trajectory tracking control problem for FJRs

3.2.1 Trajectory tracking problem:

Given a smooth reference trajectory for the link’s position , to design the input for the pH-FJR (39) such that the link’s position converges asymptotically/exponentially to the reference trajectory , as and all closed-loop system’s trajectories are bounded.

3.2.2 Proposed solution:

Using the v-CBC method in Section 2.1.4, design a control scheme with the following structure:

(41)

where the feedforward-like term ensures that the closed-loop virtual system has the desired trajectory as steady-state solution, and the feedback action enforces the closed-loop virtual system to be differentially passive.

4 Trajectory-tracking control design and convergence analysis

Before presenting our main contribution, we recall a v-CBC scheme for a fully actuated rigid robot manipulators21 with -dof, which will be used in the main result. To this end, we assume that this rigid robot is modeled as the pH system (27), describing the links dynamics only. In order to avoid notation inconsistency between the rigid and flexible controllers, this is stressed by adding the subscript to its state and parameters in (27), i.e., and , respectively.

[Reyes-Báez 19] Consider the links dynamics given by (27) and its associated virtual system (31). Suppose that and let be a smooth reference trajectory. Let us introduce the following error coordinates

(42)

where the auxiliary momentum reference is given by

(43)

with666The term is written explicitly in (42) just for sake of clarity in the following developments. , function is such that ; and a positive definite Riemannian metric tensor satisfying the inequality

(44)

with , uniformly. Consider that the -parametrized composite control law given by

(45)

with

(46)

where the -th row of is a conservative vector field777This ensures that the integral in (46) is well defined and independent of the path connecting and ., and is an external input. Then, system (31) in closed-loop with (45) is strictly differentially passive from to