The output-based control of a redundant robotic manipulator with relevant and adjustable joint stiffness is addressed. The proposed controller is configured as a cascade system that allows the decoupling of the actuators dynamics from the arm dynamics and the consequent reduction of the order of the manipulator dynamic model. Moreover, the proposed controller does not require the knowledge of the whole robot state: only the positions of the actuators and of the joints are necessary. This approach represents a significant simplification with respect to previously proposed state feedback techniques. The problem of controlling simultaneously the position trajectory and the desired stiffness in both the joint and work space is investigated, and the relations between the manipulator redundancy and the selection of both the joint and work space stiffness of the manipulator are discussed. The effectiveness of the proposed approach is verified by simulations of a 3 degrees of freedom planar manipulator.

Physical human-robot interaction currently represents one of the most challenging issues in robotics. To this end, a new class of robots is under development considering safety and dependability as a primary goal. While early applications of the variable stiffness concept were developed for prosthetic purposes due to the bio-inspired ability of these actuation systems of emulating the behavior of human muscles [

Although several prototypes of single degree-of-freedom (DOF) joints with relevant and adjustable compliance for robotic manipulators have been designed and developed [

In this paper, the control of robotic manipulators with variable joint stiffness is addressed in the general case of a redundant arm by means of a simplified (with respect to previous ones) output-feedback control approach. The proposed control is configured as a cascade system composed by three independent controllers, one for each part in which the dynamics of the robotic arm with variable stiffness joint can be subdivided. A singular perturbation analysis is then applied to the resulting system to show the separation of the dynamics of the different components, that are the arm, the positioning actuators, and the stiffness actuators, obtaining in this way a reduced order model of the manipulator and then a simplified control structure. The proposed control approach is then validated by means of simulations of a three-link planar manipulator with variable stiffness actuation. In this paper, we refer to a variable stiffness joint as a robotic joint actuated by means of a variable stiffness actuator.

Starting from the general dynamic model of a robot with

A scheme of the working principle of a variable stiffness joint is shown in Figure

Working principle scheme of a variable stiffness joint.

where

where

as expressed by (

As a nonrestrictive assumption, all the effects due to frictions, dead zones, nonmodeled dynamics, parameters variations, and so forth, not considered in other terms of the system dynamics (

As stated before, the overall system is composed by

while it is assumed that only the state subset information

are directly measurable, typically by means of optical encoders. Note that the joint stiffness

In any case, the control objective is the simultaneous and independent regulation of the following set of outputs:

namely, the joint positions (and thus, through the robot direct kinematics, the end-effector pose) and the joint stiffness, acting on the system control inputs

by means of an output-feedback controller based on the knowledge of

In the problem considered here, the main control goal is to achieve the tracking of suitable workspace trajectories assigned to the manipulator end-effector while maintaining a desired level of

The term

An important point to note is that in case of an unpredictable impact, only the mechanical stiffness of the manipulator is experienced by the environment (or the human) during the very short time interval in which the energy transfer due to the impact occurs [

Recalling that variable stiffness actuation has been introduced for safety purposes, and that its main role is to absorb at least part of the kinetic energy of the actuation system (i.e., usually predominant with respect to the one of the arm structure because of the high reduction ratio gearboxes used to connect the actuators to the links) during unpredictable impacts, another important point to note is that the selection of suitable values of the joint stiffness depends mainly on the task to be performed (in particular on the required velocities), on both the manipulator and the actuators inertia, that together with the task specifications determines the overall system kinetic energy, and, the most important, on the criteria used to evaluate the robot safety. To this end, the definition of suitable safety criteria for robotic applications represents a key topic. While in early investigations these criteria have been inherited from automotive industries, it has been successively shown that they are not completely suitable for robotic applications because of the very different scenario [

Scheme of the overall output-based controller.

Consider the generic second-order dynamic system

where

Due to the general separation principle introduced in [

The system (

By defining the state vector composed by the system tracking error

At first, the autonomous linear system obtained by neglecting

Due to the linearity of the nominal system under the effect of the proposed controller, the origin is the unique exponentially-stable equilibrium point of the autonomous (nominal) system and the rate of convergence can be made arbitrarily fast.

Now, the implications of the previous hypothesis (

The system (

Theorem

Due to the structure of

Note that (

Generally speaking, by applying a singular perturbation analysis, from Property

The control system (

By redefining the control action in (

It is important to note that both (

and

In this section, the problem of defining the desired trajectories for the actuators given the manipulator trajectories, both in terms of position and stiffness, is addressed. Note that in case the desired manipulator stiffness is defined in the joint space only, the stiffness actuator reference signals are fully defined by

It is important to note that once the desired joint stiffness

where

where

Let us define the direct kinematic problem and the manipulator Jacobian as

where

If the manipulator trajectory is defined in the task space, the vectors

In the following discussion, the leading superscript

Several works can be found in the literature about the workspace stiffness control of both redundant and nonredundant robots. In particular, in [

The relation between the workspace stiffness

It is now important to remark that the introduction of variable stiffness joints is useful to reduce the energy transferred from the manipulator to the environment (or humans) during unpredictable impacts by decoupling the actuators inertia from the link during fast robot motions (i.e., by reducing the joint stiffness) or, on the other hand, to increase the precision of the manipulator during slow and limited movements by imposing high joint stiffness values to allow adequate external disturbance force/torque rejection. Moreover, variable stiffness actuators are usually designed to have high compliance in rest conditions and to become rigid if necessary under the action of the control system. In this sense, it is then clear that the role of the joint mechanical stiffness, or

By introducing the concept of combined compliance control reported in [

The contribution to the workspace stiffness due to the mechanical joint stiffness

It is important to note that (

On the basis of these considerations, the workspace stiffness

where

From (

Due to the effect of the joint apparent stiffness, the relation between the infinitesimal joint displacements and the joint torques can be expressed as

Taking into consideration the relation between the workspace displacement and joint displacement, by inverting (

where the relation between the workspace stiffness and the joint apparent stiffness

where

Several robust controllers for robotic arms have been presented in the literature; see for example, [

First of all, aiming to clarify the role of the passive and the active stiffness, the static relation between the environmental force

By substituting (

Looking at the right hand side of (

With the aim of controlling the workspace motion of the manipulator, a specialized version of the controller (

It is initially assumed that the force

By recalling Property

In case a direct measure of the force applied by the end effector to the environment is not available, (

This equation defines a

The last term in (

Once the torque

An important remark is that while the workspace stiffness

Note that also the arm controller provides a filtered estimation of the disturbance acting on the joints through (

With respect to the control approaches for the system (

The complexity of the controller is significantly reduced since neither static nor dynamic inversion of the system is needed.

Redundancy of the manipulator is considered.

The computation of the Lie derivative of the system dynamics up to the fourth order is avoided.

An output-feedback control law has been defined instead of state-feedback controllers.

The arm trajectory must now be defined only up to the second order derivative, as in the rigid robot case, while previously, it was necessary to define the position trajectory up to the fourth order and the stiffness trajectory to the second order derivative.

Since the arm control is based on the two signals

The joint stiffness is controlled by a specific control system that directly compensates for the effects of disturbances, external forces and couplings due to both positioning actuators and arm dynamics.

The proposed control strategy exploits the mechanical compliance of the transmission to achieve the desired workspace behavior (passive compliance).

In case of perfect knowledge of the arm dynamics, the system is able to estimate and regulate the workspace interaction force without a direct measurement of the force itself.

The manipulator

The extension of the proposed approach to the case of robotic manipulator where the stiffness can be adjusted only for some joints is straightforward, since by applying the proposed positioning and stiffness actuators controller where necessary, the overall robot dynamics is reduced to the rigid case.

The proposed controller has been validated by means of simulations of a planar three-link robotic arm with variable joint stiffness. Only the Cartesian coordinates

This stiffness dynamic model [

Nominal parameters of the three-link planar manipulator.

Description | Value | Unit |
---|---|---|

Joint inertia | 1.15 | Kg m |

Joint viscous friction coeff. | 0.001 | N s m |

Link mass | 0.541 | kg |

Link center of mass | 0.085 | m |

Link length | 0.3 | m |

Motors inertia | 6.6 | kg m |

Motors viscous friction coeff. | 0.00462 | N s m |

Actuators Coulomb friction level | 0.1 | N m |

Joints Coulomb friction level | 0.5 | N m |

2.2 | ||

0.4 | N | |

14.7 | N m rad |

Parameters of the three-link planar manipulator controller used in the simulations.

Symbol | Case no. 1 | Case no. 2, no. 3 |
---|---|---|

diag | diag | |

diag | diag | |

diag | diag | |

90 | 90 | |

2.7 | 2.7 | |

10 | 10 |

parameters of the positioning and stiffness actuators fast controllers used in the simulations.

Symbol | Case no. 1, no. 2, no. 3 |
---|---|

1 | |

150 | |

9 | |

2.7 | |

1 |

Response of the system during test no. 1.

In Figure

Response of the system during test no. 2.

For the simulation of case no. 3, whose results are reported in Figure

Response of the system during test no. 3.

In this paper, the problem of output-based regulation of robotic manipulators with variable stiffness actuation has been discussed. A control scheme has been proposed whose main features are the simultaneous and decoupled workspace stiffness-position control with limited error, disturbance compensation and estimation-control of the environment interaction force. Moreover, the case of manipulator redundancy has been explicitly addressed and some nice features and implications related to the control of the both the joint mechanical stiffness and workspace stiffness have been highlighted. The control objective is achieved by means of a multilevel controller that decouples the dynamics of both the positioning and the stiffness actuators from the arm dynamics. This result is obtained by applying a singular perturbation analysis to both the actuators and the arm dynamics, configuring in this way a multilevel control structure that presents fast controllers for the actuators and a slow controller for the arm. Moreover, the proposed controller presents several advantages with respect to previous approaches for this class of devices, as the rigid-robot-like trajectory specification, the reduced complexity and the availability of the system residuals for collision detection/reaction purposes to cite some of them. The extension of the proposed analysis to an hybrid case, in which both rigid and elastic joints are present, with constant or variable stiffness, can be easily achieved. Also, the case of joints with different stiffness variation models can be easily considered. The proposed control approach is validated by means of simulation of a three-link planar manipulator, and the experimental validation of the proposed approach will be considered in the future.

The general second-order dynamic system (

In case of (

In case of (

In case of (

The general second-order dynamic system (

Let

Let consider (

Now (

This research has been partially supported by the EC Seventh Framework Programme (FP7) under Grant agreement no. 216239 as part of the IP DEXMART (DEXterous and autonomous dual-arm/hand robotic manipulation with sMART sensory-motor skills: a bridge from natural to artificial cognition) and by the Italian MIUR with the PRIN2007CCRNFA-004/PRIN2008-ZRLPWS projects.