CC BY
0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 599-612. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231203
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70E55
Design of a Robotic Spherical Wrist with Variable
Stiffness
A. A. Demian, A. S. Klimchik
This paper discusses the design of an adjustable force compensator for a spherical wrist dedicated to robot milling and incremental sheet metal forming applications. The design of the compensator is modular and can be introduced to any existing manipulator design as a single multi-body auxiliary system connected with simple mechanical transmission mechanisms to the actuators. The paper considers the design of the compensator as an arrangement of elastic springs mounted on moving pivots. The moving pivots are responsible for adjusting the stiffness of the wrist-compensator coupling. Special attention is given to two compensation schemes in which the value of the external force can be known or unknown, respectively. The simulation results show that the analytical derivation of the compensator leads the main actuators to spend zero effort to support the external force.
Keywords: static balancing, force compensation, manipulator design, variable stiffness
1. Introduction
Industrial manipulators find application in a wide range of tasks due to their versatility and precise performance. Nonetheless, one notable drawback of manipulators is their bulky linkages, which consume a significant amount of operational energy [1, 8]. Consequently, this limits the payload capacity of serial manipulators. Furthermore, in robotics applications such as milling
Received September 07, 2023 Accepted November 22, 2023
This work was supported by the Russian Science Foundation (Project number 22-41-02006).
Albert A. Demian a.demian@innopolis.university
Center for Technologies in Robotics and Mechatronics Components, Innopolis University ul. Universitetskaya 1, Innopolis, 420500 Russia
Alexandr S. Klimchik AKlimchik@lincoln.ac.uk
Lincoln Centre for Autonomous Systems (L-CAS), School of Computer Science, College of Science, University of Lincoln
Brayford Pool, Lincoln, Lincolnshire, LN6 7TS UK
and incremental sheet metal forming, manipulators are exposed to high external loads. Besides, well-constructed machine tools are nearly 100 times stiffer than a robot manipulator [33]. This results in positional inaccuracies due to the inherent flexibility of mechanical components [25, 26]. To address these issues and ensure precise operation, mechanical compensation is needed.
To achieve reliable compensation, incorporating lightweight elastic elements can be an effective strategy without adding excessive weight to the robot, thus decreasing the required torque for operation. The concept of utilizing springs for compensation was initially introduced in spring-based gravity compensators, which aimed to counteract the influence of the manipulator's link weights on its joints. It is worth noting that the integration of compensatory mechanisms has a significant impact on the manipulator's behavior and leads to nonlinear changes in its stiffness properties [23].
Besides mechanical compensation, alternative software-based solutions also exist to address the compensation challenge. Some of these solutions rely on pure stiffness modeling, while others utilize real-time feedback from primary and secondary encoders [17, 24]. However, it is important to emphasize that this paper focuses primarily on the mechanical compensation aspect.
In milling applications, interaction force is specified by tool geometry, feed rate, and rotation speed. In the case of milling, conditions do not change, and the force can be treated as constant with some variation around the mean value. Some experimental and mathematical models for the interaction force estimation in milling applications are given in [34, 35]. In [36] the authors presented an experimental study on the accuracy improvement for robot machining. The paper analyzes the estimation of milling forces and the evaluation of manipulator stiffness for the configuration considered, including the influence of the gravity compensator on the manipulator's behavior under loading.
The issue of achieving gravitational balance at a mechanical level has undergone extensive research over several decades. Various methods of compensating for gravity are outlined in [1, 4, 7, 10, 14]. This approach involves creating a model of the manipulator's potential energy and configuring the springs appropriately. By optimizing the spring arrangement, the variation in potential energy within the manipulator's workspace is minimized, resulting in reduced joint torques. [2] provides an overview of gravity balancing mechanisms for both single- and multi-degree-of-freedom systems. Since gravity torques are influenced by the mass properties of the robot's components, they are treated as a modeled parameter in the context of mechanical gravity compensators. This paper explores a similar concept that can handle unforeseen loads the robot may encounter during its operation.
Variable stiffness actuators (VSAs) have the goal of modifying the stiffness of the actuator to alter the range of torque it can generate. Various types of VSAs are detailed in [28-31], where multiple approaches and mechanisms are presented. The underlying idea behind variable stiffness actuation involves integrating actively adjustable elastic components with the actuator's shaft. Adjusting the tension in these elastic elements leads to changes in the actuator's torque output. It is worth noting that the primary purpose of VSAs is to ensure safe interactions between humans and robots, particularly in scenarios where collisions are likely to occur. In simpler terms, VSAs are designed to change their position when subjected to increased external loads. Furthermore, the concept of VSAs encompasses methods for adjusting stiffness at a localized level, focusing on individual joints.
This study addresses the issue of compensating for external forces in mechanical design, considering variations in both the direction and magnitude of the force. In contrast to Variable Stiffness Actuators (VSAs) which act locally at a single joint, the described force compensator aims to compensate for the compound effect of the external force by considering the relative
position of different joints. The goal is to enhance positional precision when faced with changes in external loads. Similar to mechanical gravity compensators, the fundamental idea involves utilizing the adjustment of linear springs to control the stiffness of the overall structure, thereby reducing the actuator torques necessary to support the external load. The core principles of this design and the derivation process can be found in [32].
2. Compensator for a 2-DoF RR system
Fig. 1. A geometric representation of a spherical wrist
This section presents a detailed derivation of the compensator parameters. The considered kinematic configuration of a ZYZ spherical wrist with an arbitrary external force acting at the end effector is shown in Fig. 1. The system's kinematics can be derived as follows:
0T2(9i, q2, 9s) = Rz(Qi)Transz(ll)Ry(q2)Rzq)Transz(l2),
(2.1)
where qi denotes the local joint coordinates, li denotes link lengths, Rx y z is a 4 x 4 homogeneous transformation matrix of rotation around the subscripted axis, and Trans denotes a homogeneous transformation matrix of translation along the subscripted matrix.
The end effector position is the first three elements in the last column of 0T2:
JX0
l2 sin(q2)cos(qi) l2 sin(9i)sin(92) ll + l2 COS(92)
(2.2)
where X denotes the position vector of the end effector, l1 is the distance from the wrist's base (frame #0) to the center of the wrist (frame #1), l2 is the distance from the wrist's base to the end effector (frame #2), and [q1, q2] are the generalized local joints coordinates.
In order to be able to work with Eq. (2.2), transformation using trigonometric identities is needed to transform the multiplication of trigonometric functions into a sum:
f [sin{q1+q2) - sm{q1 - q2)\ |[cos{ql - q2) - cos{ql + q2)\
l1 + l2 cosfe)
(2.3)
This transformation will allow representing all the trigonometric functions in sum or difference of angles which can be realized with a simple mechanical transmission.
The static potential energy of the manipulator against the external force can be modeled as follows:
Um — °X2 F
fx l2
[sin(qi + q2) - sin(qi - ^2)] + fyl2
+ -^-[cos(i! - q2) - cos(i! + q2)} + fz{l\ + l2cos{q2)). (2.4)
2
Fig. 2. A geometric representation of a unit compensator with two coordinate angles
The system in Fig. 2 shows a mechanism representing a trigonometric function with a sum or difference of angles. The purpose of the spring mounted between both tip points is to create counter torque. The coordinates of the mounting points are the following:
A — acos(a1) , B — bcos(a2)
a sin(a1) b sin(a2)
(2.5)
Considering a zero free-length spring with stiffness coefficient k, the potential energy of the system is formulated as follows:
U = ^k\\B-A\
= ^k [a'f + b\ — '2a,1b1 cos^ — a2)),
(2.6)
where k is the stiffness coefficient of the spring and U is the potential energy.
One can notice in Eq. (2.3) the existence of 5 distinct trigonometric functions. Therefore, it is possible to realize each function with different configurations of the mechanism shown in Fig. 2. The angular coordinate of the spring attachments is a function of manipulator coordinates. One of each spring attachment points will be of variable length, while the second will be of fixed length.
2
The number of the adaptive parameters corresponds to the effective external force components. Since Eq. (2.4) shows three active force components, three adaptive parameters are placed:
A,
A0
Ao
a1 cos(q1) a1 sin(q1)
B
11
a2 cos
9 +
a2 sin
0
h + 2)
B
21
B
31
bn cosfe) bii sin(q2) b2i cosfe) b21 sin(?2)
b31 cos(^2) b31 sin(?2)
B
12
B
22
b12 cos(^ - 92) b12 sin(^ - 92)
b22 cos(n - 92) b22 sin(n - 92)
(2.7)
(2.8) (2.9)
where Ai is the position vector of a sliding pivot and Bij is the position vector of the attachment point of the spring connected to point Ai.
A geometric representation of the compensator is shown in Fig. 3a. The total elastic potential energy of the compensator will be modeled as follows:
1 m n i=1 j=1
= -fciiih&n cos(q1 - q2) + (a? + b2n) + fc12a.ibi2 cos(q1 + q2) + ^k12 (a? + b\2) + + fc2la2b21 s[n(qi - q-2) + \k2l (a2 + b2l) - k22a2b22 sin(<?l + Q2) + \k22 (a2 + b22) ~
- hia-ihi cos(ç2) + (a| + (2.10)
where m is the number of the attachment points Ai and n is the number of the attachment points Bij which are connected to point Ai through a spring with a stiffness coefficient kij.
(a) A scheme of the compensator design with indication of the pivot points
(b) A scheme of wrist-compensator coupling
Fig. 3. A geometric representation of a unit compensator with two coordinate angles
When the compensator and the wrist models are mechanically coupled together as shown in Fig. 3(b), the total energy of the system can be counted as the sum of manipulator and elastic
2
a
3
energies:
Utotal = Um + Ue. (2.11)
Partial derivatives with respect to joint coordinates provide the values of joint torques to hold the force in equilibrium:
dUtotai = (knaibn _ Ml) s]n{qi _ g2) + k12aib12) sin(9l + q2) +
dq1 y'"11^11 2 ) V 2
/xl2\ „„„/„ „ \ , I fxl2
+ [k21a2b21 -~y-J cos(qi - q2) + - k22a2b22 ) cos(9l + q2), (2.12)
dUtotai ffv12 _ sin(9i _ q2) + (Ùà _ kl2alhv2 ) sin((?1 + g2) +
dq2 V 2 -^-^^j—v*1 ^ ■ ^ 2
+ (J^ ~ k2la,2b2^j cos(i! - q2) + (J^ - k22a2b22^ cos^ + g2)+
+ (k31 a3b31 - fl2) sin(q2). (2.13)
It is desired that the joint torques be brought to zero. One simple solution is to find the values of a1, a2, and a3 that will bring all coefficients of the trigonometric functions to zero. Thus, analyzing the derivatives will yield the following solutions for parameters a^:
«2 = fx-^rj- = fxj\ , (2-!5)
2k21b21 2k22 b22
k31b31
Since parameters a1 and a2 have multiple solutions in Eqs. (2.14) and (2.15), a constant ratio between solutions should be found for them to be equal:
fcn = ^ (2-17)
_ k2
J21
b21 = (2.18) k21
It is worth noting that in Eqs. (2.14), (2.15), and (2.16) each parameter, a1, a2 or a3, is directly co-related to a force component fx, fy or fz, respectively. Accordingly, this means that the compensator decouples the effect of the force components and is able to compensate for any arbitrary force. Moreover, the solution presented in Eqs. (2.14), (2.15), and (2.16) brings the joint torque required to support the force to zero independently of the joint configuration.
3. Force transformation into the wrist base coordinate frame
The previous chapter explained the synthesis of the compensator that compensates for an externally applied force in Cartesian space with respect to the coordinate frame of the wrist's base. A typical use of a spherical wrist is to be mounted on a robot elbow, hence, the orientation of the wrist's base frame is constantly changing. Therefore, considering the external force to be
represented in the base frame of the wrist is insufficient for the compensation to be effective. Frame transformation is needed to correctly represent the force in the wrist's frame.
Let us consider a manipulator with n revolute joints and 3 joints in the spherical wrist n + (1, 2, 3). The orientation kinematics of the end effector can be modeled as follows:
Re — R
n+1 R
n+1 -"e
(3.1)
where 0Rn is the orientation matrix from the global base frame to the joint n of the manipulator and n+1Re is the rotation matrix of the spherical wrist.
The orientation of the wrist can be extracted by premultiplying by the inverse of the arm's rotation matrix. Since the columns of the rotation matrix form an orthonormal set, the transpose of the matrix is equivalent to the inverse R_1 = RT:
n+1 R —Or, T 0B Re _ Rn+1 Re.
(3.2)
Accordingly, to represent the external force in the frame of the spherical wrist, the force vector should be similarly transformed:
(3-3)
F _O rT f
where Fw is the vector of the external force represented in the wrist's frame.
Equations (2.14) (2.15), and (2.16) show the mapping between the force components and the adaptive parameters. This mapping assumes that the wrist base is stationary:
a2
a1 —
a3
0
0 0
0
fx
fy
fz
(3.4)
According to Eq. (3.3), the adaptive parameters considering the variable orientation of the wrist's base can be modeled through substitution into Eq. (3.4):
a2
a1 —
a3
0
0 0
0
Rn+1 (q0> ... In)
T
3x3
fx
fy fz
(3.5)
This mapping takes in consideration the moving base frame of the spherical wrist while ensuring that the compensation for the external force is independent of the configuration of the wrist's joints.
2
2k21 b21
2
2k11b11
2
k b
2
21 b21
2
2k11 b11
k31b31
4. Compensation based on internal torque feedback
This section aims to present the adaptation law in case the value of the external force is unknown. The previous section explained how the force value changes for the wrist given that the wrist's base frame changes as a function of the position of the preceding manipulator's joints. Since the relationship between the adaptive pivots and the external force has been modeled in the previous two sections, an adaptation law can be derived, while the value of the external force is unknown given the kinematics of the manipulator and the value of the torque from the internal feedback of the joints.
The torque values of the wrist's joints that are needed to support the external force are presented in Eqs. (2.12) and (2.13). The adaptive parameters are responsible for compensating for the effect of the external force and bringing the torque value to zero, which in turn makes the desired value of the torque Td — 0. Hence, the torque error can be modeled as follows:
ST — Td - Tr, (4.1)
where Td is the desired torque value and Tr is the real torque value.
Since the external force and the joint positions are the same in both Td and Tr, the change in the torque only depends on the variable parameters a1, a2 and a3:
St — f (a1, a2, a:i). (4.2)
To derive the mapping relationship between the adaptive parameter and the error in torque, the partial derivative with respect to the parameters ai can provide the following relationship:
ÔT = JJA = ÏÏÂ
au'
dq1
du -dq2_
(4.3)
where Ja denotes a mapping matrix between joint torque and the adaptive parameters and A is the vector of adaptive parameters. This yields the following matrix:
Ja
k12b12(sin(<?1 - Q2) - sin(<?1 + Q2) k22b22(c0s(<?1 - ?2) - C0s(<?1 + ?2) 0
-^12^12(sin(01 - Q2) - Sin(?1 + Q2) -k22&22(C0s(^1 - ^2) - C0s(^1 + ^ ^31^31 sin(^2)
(4.4)
The change of the adaptive parameters can be calculated using Eq. (4.3):
S A — J-1 ST. (4.5)
However, since the Ja matrix is not a square matrix, the direct inverse is not calculable. Alternatively, the pseudo-inverse will be calculated instead:
SA — J Ja)'1 JTa ST. (4.6)
This mapping provides a method to compensate for an unknown external force. It is worth mentioning that this method can be performed with knowledge of information about the spherical wrist only. Information about manipulator kinematics can be considered irrelevant to compensate for the torques in the wrist. However, knowledge of the joint positions and torques in the wrist's joints is necessary.
5. Discussion
This paper presents a method for compensating joint torques in a spherical wrist that are induced by an externally applied force. The compensator can be realized as a mechanical auxiliary system that can be mechanically coupled with the wrist's joints. This implies the fact that this method can be realized for the typical design of manipulators without the need to redesign the manipulator's joints. Mechanical springs enhance energy efficiency: they are lightweight
while providing high elastic forces support for the main manipulator. The design of the compensator presented is parametric and can be synthesized for a variety of dimensions of a spherical wrist. Coupling of the compensator with the wrist can be performed using a simple mechanical transmission, a gear mechanism or a pulley-belt arrangement since it relies on transmitting the 1 : 1 ratio of joint positions. It is noticeable in the particular case of the spherical wrist that the position of the third joint is irrelevant since its actuation axis lies along the line of effect of the force. However, if the force position shifts, the design of the compensator needs to be reconsidered since the current design considers that the point of effect of the force lies along the actuation axis of the third joint.
The positioning of the adaptive parameters in the compensator is chosen so that the adaptation is performed in direct proportion to the external force. Although Eqs. (2.12) and (2.13) make a redundant system of equations in three variable parameters a1, a2, and a3 considering that the joint position and the external force are constant, the aim is to find a solution that makes the adaptation independent of the joints' position. That is, by solving all the coefficients of the trigonometric functions equal to zero, the effect of the force on the joint torques will be canceled. Hence, all the efforts spent by the joints are control and gravity efforts. This is useful since in milling applications and sheet metal forming the reaction force from the spring effect of the material is far larger than the gravity of control efforts. Also, it should be noted that the joints or the robot wrist are smaller than those of the elbow.
The adaptive parameters are responsible for changing the springs' configuration in the compensator. This implies a change in the stiffness of the whole wrist. Accordingly, it is possible to consider that the adjustment of the adaptive pivots is directly controlling the stiffness properties of the wrist. In a typical spring, the applied force deforms the spring to an equilibrium point where the elastic force equals the applied force, thus holding the force constant at equilibrium. On the contrary, the compensator aims to adjust the stiffness of the rest so that the current position of the wrist becomes the equilibrium position for the applied force. Provided that the external force changes in value, the adaptive parameters aim to change the stiffness of the wrist accordingly to prevent positional deflections.
Since the path planning process in milling is usually performed offline, it is possible to have an estimation of the milling force along the planned path. A hybrid offline-online compensation scheme can be developed based on the proposed system. In this case, the offline compensation model is responsible for the mechanical compensator's behavior, while the online compensation model is used to compensate variations from the mean estimated values.
6. Results
The model is tested computationally in two scenarios in which joint torques are calculated based on the state of the wrist-compensator coupling. Tests were computed over 2000 iterations where an iteration is a dimensionless step unit. One scenario considers the wrist's base to be stationary, while the wrist joints are moving in a path. The second scenario considered the change in orientation of the spherical wrist while the wrist's joints were stationary. In both scenarios, the value of the external force is constant in the Cartesian frame. The wrist and compensator parameters were assigned according to Table 1.
The first test was performed considering the base of the spherical wrist is stationary, hence, the external force vector is constant in the base frame of the wrist. A force vector of value [12, 8, 10]^ is applied at the end effector while the wrist's joints are moving. The values of
Table 1. Manipulator and compensator parameters used in the simulation
manipulator parameters compensator parameters
parameter unit value parameter unit value
h 111111 400 bn, b12 111111 150
h 200 ^21> ^22) &31
k12 133
Ql, <72, <fe rad 7r 7T I L~2' 2.1 k21, k22 N/111 133
^31 180
Potential energy of the system
Joints position of the wrist
12 IT 10 â s fc 6 4
~ ^elastic ^force
y
Utotal ---- N, ^____ >• - ✓ -s. /
N N V A' ___' * ^ X
-1- - --1---
M
a
cS
0 250 500 750 1000 1250 1500 1750 2000 0 500 10001500 2000 Torque an counter-torque at the first joint q1
2
? 1
è 0 -2
N
^elastic \
^force . Tcompensated
0 250 500 750 1000 1250 1500 Torque an counter-torque at the second joint q2
1750
2000
2000
Fig. 4. Simulation of potential energy and joint torques considering stationary wrist's base and subjected to external force vector of F = [12, 8, 10](W)
the adaptive parameters calculated using Eq. (3.4) are constant since the force vector is constant. The results of the simulation in Fig. 4 show that the potential energy of the wrist-compensator coupling is constant for a constant force. This implies that the energy is conserved in the system without the need to introduce any energy to support the external force. As a result, the compensated actuator torque is zero, which indicates that the external force is supported by the compensator.
The second test was performed considering that the base of the spherical wrist is moving with an arbitrary manipulator elbow. Hence, the external force vector is changing in the base frame
of the wrist. A force vector of value [12, 8, 10]^ is applied at the end effector while the wrist's joints are kept stationary. The values of the adaptive parameters calculated using Eq. (3.5) are updated at each step since the orientation of the wrist is changing. The results of the simulation in Fig. 5 show that the potential energy of the wrist-compensator coupling is changing, while the compensated torque of the joints is kept at zero. This implies that the actuators do not spend energy to support the external force, while the adjustment of the adaptive parameters a1, a2 and a3 requires spending energy to change the stiffness parameters of the wrist-compensator coupling.
Potential energy of the system
Orientation of the wrist's base
15
Tio
à 5
^elastic ^force
---- ---- ----
■«■ «»
0 250 500 750 1000125015001750 2000
Torque an counter-torque at the first joint q1 2-
1.0
i^-0.8 'V
¿0.6 JH 0 4 § 0.2 0.0
■>
- Rx
Ry
—
0 250 500 750 10001250 1500 1750 2000 Transformed force to the frame of the wrist 15
0 250 500 750 1000125015001750 2000 Torque an counter-torque at the second joint q2
0 250 500 750 10001250 1500 1750 2000 The linear position of the adaptive pivots
0 250 500 750 1000125015001750 2000 Iteration
0 250 500 750 10001250 1500 1750 2000 Iteration
Fig. 5. Simulation of potential energy and joint torques considering the motion of the wrist's base and subjected to external force vector of F = [12, 8, 10](N)
7. Conclusion
In this paper, we have discussed the design of an adjustable force compensator for a spherical wrist which aims to compensate for an externally applied force. The moving pivots of the compensator are responsible for adjusting the configuration of the elastic springs, which results in a change of the stiffness properties of the wrist-compensator coupling. The simulation shows that the potential energy of the system is constant, while the force value is constant and the base of the wrist is stationary. However, the potential energy of the system changes accordingly
as the orientation of the base of the wrist changes since it corresponds to changes in the force vector. In either case, the compensator manages to keep the torques of the main actuators at zero. The adaptive pivots of the compensator are responsible for compensation for the change in the potential energy since it corresponds to adjusting the stiffness of the system. This enhances the performance of the main actuators and allows them to be used for position control instead of wasting energy supporting external loading. Nevertheless, the main actuators are still required to exert control and dynamic efforts which are disregarded in this paper.
Conflict of interest
The authors declare that they have no conflict of interest.
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