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Advances in Mechanical Engineering
Volume 2010 (2010), Article ID 284976, 9 pages
Application of the Rotation Matrix Natural Invariants to Impedance Control of Rotational Parallel Robots
1Dipartimento di Meccanica e Costruzione delle Macchine, Università di Genova, Via Opera Pia 15A, 16145 Genova, Italy
2Dipartimento di Meccanica, Università Politecnica delle Marche, Via Brecce Bianche 12, 60131 Ancona, Italy
Received 17 June 2009; Accepted 10 November 2009
Academic Editor: Zhen Huang
Copyright © 2010 L. Bruzzone and M. Callegari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Force control of parallel robots with rotational degrees of freedom through impedance algorithms is considerably influenced by the representation method of the end-effector orientation. Using the natural invariants of the rotation matrix and the angular velocity vector in the impedance control law has some theoretical advantages, which derive from the Euclidean-geometric meaning of these entities. These benefits are particularly evident in case of robotic architectures with three rotational degrees of freedom (serial or parallel wrists with spherical motion). The behaviour of a 3-CPU parallel robot controlled by an impedance algorithm based on this concepts is assessed through multibody simulations, and the results confirm the effectiveness of the proposed approach.
Several robotic applications involve the control of the interaction forces between end-effector and external environment; the control laws that can fulfil this requirement can be divided into two families: hybrid position/force control algorithms and impedance control algorithms. These two approaches have been widely discussed and compared in the scientific literature during the last decades, in particular with reference to serial kinematics [1–7].
In the last years the industrial interest about parallel robots (parallel kinematics machines, PKMs) is growing; mass-produced parallel manipulators are widely used especially in two categories of applications:(i) fast speed pick-and-place tasks with small payloads; in this case high-speed, lightweight parallel robots are adopted for their excellent dynamic performances, due to the low moving masses (e.g., the FlexPicker IRB340 by ABB based on the Delta kinematics );(ii) machining or assembly tasks characterized by high forces, which requires high structural stiffness and positioning accuracy; in this case high-stiffness PKMs (e.g., the F-200iB by Fanuc based on the Gough-Stewart kinematics [9, 10] or the Tricept by Neos Robotics AB) ) offer a high ratio between maximum payload/force and robot mass.
In most cases, parallel robots are position-controlled; nevertheless, the scientific and industrial interest about force-controlled PKMs is growing, even if only a few works on the subject are presently available, probably due to the fact that PKMs are relatively new architectures, characterised by quite complex kinematic and dynamic models [12–14]. In particular, the use of impedance-controlled parallel robots [15–18] is an effective solution: the moving masses of PKMs are limited, because the actuators are usually fixed to the ground, and then it is possible to realise the desired stiffness-damping behaviour neglecting the compensations of the inertial terms.
One of the major problems in the synthesis of a suitable impedance control law (both for serial and parallel robots) is the definition of the rotational stiffness, which is strictly related to the representation method for the end-effector orientation . While the position of a point in space can be easily defined by a three-dimensional vector, there are many mathematical tools to represent the orientation of a rigid body in space [20, 21] and this influences the resulting impedance control law. In the following of the paper this aspect will be discussed with reference to simulation results involving an impedance-controlled 3-CPU parallel robotic wrist.
2. Representation of the End-Effector Orientation(i) there are 12 different sets of Euler angles, and the choice of a set influences arbitrarily the control algorithm;(ii) there are representation singularities (configurations defined by nonunique values of Euler angles), and in these singularity loci it is impossible to describe an arbitrary angular velocity as function of the Euler angles time derivatives.
Some researchers tried to overcome these limitations by using a modified set of Euler angles that avoid singularities  or the so-called Euler parameters, a 4-dimensional vector of invariants that are not geometric entities .
Another possible representation of the orientation is the rotation matrix, but its nine elements are not independent and an impedance algorithm based on three independent rotation matrix elements is poor of geometrical meaning.
An arbitrary rigid body rotation is uniquely defined by the axis of rotation (represented by the unit vector e) and the angle of rotation , which is the amount of rotation according to the right-hand rule; the unit vector e and the angle are the natural invariants of the rotation matrix, independent from the reference frame .
In  the use of the frame-invariant vector in the impedance control law is proposed. The main advantage of the use of this vector is related to its geometrical meaning and strict relation to the angular velocity vector :
In an impedance algorithm the vector E can be used to represent the rotation between the actual end-effector orientation and the reference orientation; in this case the maximum values of are limited. It is easy to demonstrate that if tends to zero, the angular velocity vector tends to the time derivative of E:
This is a very important propriety of the vector E, because an impedance control law based on the vector E for the stiffness term and on the angular velocity vector for the damping term uses two vectors that are nearly in relation of time derivative. This important propriety is not shared by the other orientation representation methods, and it leads to a more natural and intuitive robot behaviour. The advantages of such an approach in case of a six-degree-of-freedom parallel robot are shown in ; in case of a purely rotational three-degree-of-freedom parallel robot the benefits are even more evident, and this will be shown in the following of the paper.
3. The 3-CPU Parallel Architecture
In the 3-CPU architecture  (Figure 1) three identical serial chains (legs) connect the fixed base and the rotating end-effector platform; each leg (Figure 2) is composed of two links; the first link is connected to the base by a cylindrical joint and to the second link by a prismatic joint; the second link is connected to the end-effector by a universal joint.
The axes of the cylindrical joints , , intersect at the centre of motion and are aligned to the axes of the fixed reference frame, with its centre in . The first link of th leg is perpendicular to the axis and has a variable length thanks to the prismatic joint ; the second link is parallel to . The first axis of the universal joint is perpendicular to the plane of the th leg, while the second axis intersects the corresponding axes of the other legs at the point ; these three axes are coincident to the axes of a cartesian frame located in and attached to the end-effector. If proper manufacturing and mounting conditions are fulfilled, the centres of the fixed frame and rotating frame remain coincident during the motion, and the end-effector performs a spherical motion. In particular, in the initial configuration the three linear displacements of the cylindrical joints are equal to and the linear displacements of the prismatic joints are equal to (Figure 2). The spherical motion of the end-effector can be driven by actuating the linear displacements of the cylindrical joints; therefore, the vector of the internal coordinates is:
The direct and inverse kinematics of the robot make the forward and backward connections between the internal coordinates and the relative rotation of the frame with respect to .
The choice of the three external coordinates influences the impedance control law; in the following, two possible sets of external coordinates are considered:(i) the set of Euler angles , , corresponding to three subsequent rotations along the axes that represent the overall relative rotation between and ; (ii) the three components of the vector that represent the overall relative rotation between and .
Independently of the selected set of external coordinates, the direct kinematic problem admits up to 8 solutions, while the solution of the inverse kinematics is single . The choice of the external coordinates determines the consequent Jacobian matrix: deriving the two possible sets of inverse kinematics equations, two alternative analytical Jacobian matrices can be found:(i) the analytical Jacobian matrix , which relates the time derivative of q and the time derivatives of the Euler angles, collected in the vector : (ii) the analytical Jacobian matrix , which relates the time derivative of q and the time derivative of E:
The geometric Jacobian represents the relationship between the time derivative of q and the end-effector angular velocity vector :
4. Impedance Control Algorithm
The basic concept of impedance control is that the robot end-effector, subject to external forces, follows a trajectory with a predetermined spatial compliance [28, 29]; this compliance, which influences the end-effector dynamic behaviour, is defined by the stiffness matrix K and the damping matrix C; the reference configuration of the end-effector is a virtual equilibrium state because it corresponds to an actual equilibrium only if no force is exerted by the environment.
The actual position of the end-effector can be represented alternatively by the external coordinates vectors or ; similarly, the virtual equilibrium configuration, corresponding to the reference frame , can be represented by the Euler angles or by the vector . Using the principle of the virtual work and considering (4), (5), and (7), the two impedance control laws (8) and (10) can be derived, the first based on the Euler angles and the analytical Jacobian matrix , and the second based on the vector E and the geometric Jacobian matrix :
where :(i)F is the vector of the three actuation forces, acting along the axes of the cylindrical joints;(ii) is the rotational stiffness matrix, which expresses the relation between rotation and elastic torque in the desired impedance control behaviour;(iii) is the rotational damping matrix, which expresses the relation between rate of rotation and viscous torque in the desired impedance control behaviour.
Let us note that in the control law (10) the damping term is not based on the derivatives of the external coordinates but on the angular velocity vector. The matrices and are in general nondiagonal; it is possible to define and using a principal reference frame () in which they are diagonal; this principal reference frame can be selected on the basis of the task requirements; it is possible to obtain and in the world frame () by means of the rotation matrix :
where and are the stiffness and damping matrices expressed in the principal reference frame.
In the following section, the control laws (8) and (10) will be compared by means of multibody simulation, to show the advantages of the second control law, due to the geometric properties of the vector E.
5. Simulation Results
The impedance control laws (8) and (10) have been applied to a 3-CPU parallel robot with the geometric and inertial parameters shown in Table 1. Since the two links of each leg rotate only about the corresponding cylindrical joint axis, only the moments of inertia with respect to this direction are reported; the universal joints crosses can be considered lumped masses concentrated in the corresponding universal joint centres.
Case Study A
In the case study A, starting from the initial condition with the coordinate frames and coincident, a moment with direction and magnitude 20 Nmm is applied to the end-effector. The control system imposes an isotropic rotational behaviour, defined by the following diagonal stiffness and damping matrices: The value of the desired isotropic stiffness ( Nmm/rad) is arbitrary; the value of the isotropic damping coefficient is selected using an heuristic approach, considering each rotational direction as a linear inertia-torsional spring-torsional damper system with a damping coefficient . Therefore, the following well-known formula is used: where Ns2mm is the end-effector moment of inertia (Table 1).
The rotational behaviour of the robot about the three directions can be compared with three decoupled second-order inertia-rotational spring-rotational damper systems, characterized by the following linear differential equations:
For these three decoupled systems one has the following:(i) the inertia of Ns2mm derives from the end-effector moment of inertia increased considering approximately the leg links 1 and 2 as lumped masses placed, respectively, at a distance of and from the point ;(ii) the stiffness and damping coefficients are equal to the isotropic stiffness and damping coefficients and ;(iii) the applied moments are the three components of the moment .
Figures 3, 4 and 5 show the rotational behaviour of the robot controlled by the law (8), based on the analytical Jacobian matrix, in comparison with the three decoupled systems (15); in these graphs the end-effector orientation is represented both by the components of the vector and by the Euler angles; Figures 6, 7 and 8 refer to the rotational behaviour of the robot controlled by the law (10), based on the geometric Jacobian matrix.
Both the control laws apply the isotropic stiffness and damping expressed by the matrices (13), but only the geometric control law (10) realizes an isotropic behaviour: in final steady state the end-effector rotation is rad, that is a rotation with the same direction of the external moment and magnitude equal to the magnitude of divided by the isotropic stiffness . On the contrary, using the analytical control law (8) the final rotation is rad.
Moreover, it is possible to note that using the geometric control law (10) the time histories of the three components of are very similar to the ones of the three decoupled linear systems; this means that the rotational behaviour of the robot is very intuitive and almost decoupled in the three directions.
Case Study B
In the case study B, starting form the initial condition with the coordinate frames and coincident, a moment with direction and magnitude 20 Nmm is applied to the end-effector. Now the control system imposes a nonisotropic rotational behaviour; the stiffness and damping matrices are defined in a principal reference frame , which is rotated with respect to according to the following vector E: which corresponds to the rotation matrix: The principal rotational stiffness and damping matrices are The matrix imposes a rotational stiffness about the first principal axis much lower than about the other two axes, in order to have large rotations only in one direction and negligible rotations in the other two directions, independently of the direction of the applied moment; the diagonal values of are obtained by the formula (14) as in the previous case study. The rotational stiffness and damping matrices in the reference frame can be obtained by the equations (12).
The rotational behaviour of the robot about the three principal directions is compared with three decoupled second-order inertia-rotational spring-rotational damper systems loaded by the components of the external moment :
Figures 9, 10 and 11 show the behaviour of the robot controlled by the law (8), based on the analytical Jacobian matrix, in comparison with the three decoupled systems (19); Figures 12, 13 and 14 show the same comparison, but applying the law (10), based on the geometric Jacobian matrix. In these graphs the end-effector orientation with respect to the principal stiffness and damping reference frame is represented both by the components of the vector and by the Euler angles.
It is possible to note that using the control law (8) the rotational behaviour of the robot is remarkably different from the decoupled systems; on the contrary, using the control law (10) the time histories of the three components of the vector are very close to the ones of the three decoupled systems:
(i) the relative difference between and , after 0.5s, remains lower than 0.15% (Figure 12); this rotation direction is the most significant because it is the one with higher compliance;(ii) the relative difference between and has a peak of 9.8% at 0.2s (zoom box of Figure 13) but rapidly decreases;(iii) the relative difference between and is high only in the first second (zoom box of Figure 14).
All the relative differences tend to zero in steady state. These results show that the control law (10) imposes successfully the rotational compliance about the three principal axes, and is capable of effectively constrain the rotation direction; the robot behaviour is almost decoupled in the three rotation directions and similar to a linear three-degree-of-freedom second-order system. Let us note that compete decoupling of the robot dynamic behaviour could be obtained by introducing the model-based compensation of the inertial terms , but this increases remarkably the computational burden without significant benefits.
The vector is the product of the unit vector e representing the axis of rotation and the angle of rotation , which are the natural invariants of the rotation matrix, independent from the reference frame; this frame-invariant vector has been applied to the impedance control of a three-degree-of-freedom purely rotational parallel robot (3-CPU).
If the Euler angles are used as external coordinates, the kinematics is characterized by the analytical Jacobian matrix ; on the contrary, using the components of the vector E as external coordinates leads to the definition of the analytical Jacobian matrix , which tends to the geometric Jacobian matrix if tends to zero.
This important geometric property has relevant consequences on the behaviour of the resulting impedance control; to show that, the behaviours of the 3-CPU parallel robot controlled by two different impedance algorithms, based on the Jacobian matrices and , have been compared.
The simulation results show that only the geometric Jacobian-based impedance control law allows to impose properly isotropic and nonisotropic end-effector stiffness and damping; moreover, the end-effector rotations along the three principal stiffness/damping directions are almost decoupled and similar to linear second-order systems, without the need of the model-based compensation of the inertial terms, which is computationally heavy.
For all of these reasons, using the components of the vector E as external coordinates is suitable for all the applications in which a intuitive and decoupled control of the rotational compliance is required; for example, haptic human-machine interfaces for teleoperation of robots and control sticks of real or virtual aircrafts and helicopters; compliant tables for carrying parts in robotized assembly lines.
The dynamic performance of a robot controlled by the proposed algorithm depends on the inertial properties, on the features of the actuators, and on the stiffness and damping matrices; adopting a proper mechanical design in combination with actuators characterized by high force/torque it is possible to impose high values of control stiffness and consequently to perform high frequency motion.
Although the main objective of impedance control is to obtain rotational compliance in one or more directions, high rotational positioning accuracy can be obtained in the remaining directions by imposing high values of the corresponding elements of the stiffness matrix; obviously, in the real implementation the maximum accuracy is limited by the mechanical stiffness and by the accuracy of the sensors.
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