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6Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 817-829. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221212
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70E55
Gravity Compensation for Mechanisms with Prismatic Joints
A. A. Demian, A. S. Klimchik
This paper is devoted to the design of gravity compensators for prismatic joints. The proposed compensator depends on the suspension of linear springs together with mechanical transmission mechanisms to achieve the constant application of force along the sliding span of the joint. The use of self-locking worm gears ensures the isolation of spring forces. A constant-force mechanism is proposed to generate counterbalance force along the motion span of the prismatic joint. The constant-force mechanism is coupled with a pin-slot mechanism to transform to adjust the spring tension to counterbalance the effect of rotation of the revolute joint. Two springs were used to counterbalance the gravity torque of the revolute joint. One of the springs has a moving pin-point that is passively adjusted in proportion with the moving mass of the prismatic joint. To derive the model of the compensator, a 2-DoF system which consists of a revolute and a prismatic joint is investigated. In contrast to previous work, the proposed compensator considers the combined motion of rotation and translation. The obtained results were tested in simulation based on the dynamic model of the derived system. The simulation shows the effectiveness of the proposed compensator as it significantly reduces the effort required by the actuators to support the manipulator against gravity. The derived compensator model provides the necessary constraints on the design parameters.
Keywords: prismatic joints, static balancing, gravity compensation, manipulator design
Received September 12, 2022 Accepted November 21, 2022
This work was supported by the Russian Science Foundation (project number 22-41-02006).
Albert A. Demian [email protected] Alexander S. Klimchik [email protected]
Center for Technologies in Robotics and Mechatronics Components, Innopolis University ul. Universitetskaya 1, Innopolis, 420500 Russia
1. Introduction
Robots experience large static forces while operating in large workspaces. These static forces are mainly generated by gravity, which means that a large part of the energy spent during operation goes to support the robot's weight [1, 2].
Various approaches were proposed to compensate for gravity. A classical approach is to use counterweight as shown in Fig. 1a. This approach allows the manipulation of larger payloads, however, it increases the total potential energy of the system [3]. Other approaches are based on auxiliary mechanisms with spring suspension similar to the one in Fig. 1b. The advantage of using springs is that they are light in weight, hence, the increase in the system's potential energy is insignificant. Moreover, springs can store potential energy which reduces the energy required for operation. Many mechanisms for gravity compensation are summarized in [4]. More mechanisms are presented in detail in [5-14]. These mechanisms achieve different results between complete and partial compensation for links' weight.
electric motors used as counterweights
active balancing system
(a) Gravity compensation using counterweight
""
(b) Gravity compensation using an auxiliary mechanism
Fig. 1. Examples of gravity compensation
It should be mentioned that implementation of gravity compensators affects the manipulator's behavior as well, its stiffness properties become nonlinear [15]. Apart from the mechanical gravity compensation, there are software-based solutions which rely either on pure stiffness modeling [16] or real-time feedback from primary and secondary encoders [17].
The majority of those mechanisms address gravity compensation for revolute joints. Moreover, a limited number of studies addressed compensation for prismatic joints. Gravity compensation for prismatic joints is addressed in [18-20]. The main drawback of these proposed mechanisms is their size if the prototype is realized according to Fig. 2. A challenge regarding gravity compensation for prismatic joints is that the center of mass shifts accordingly with motion.
In this paper, we propose a preliminary concept of a gravity compensator for prismatic joints. The mechanism depends on spring tension together with a combination of pulleys and gear transmission. Moreover, this mechanism aims to compensate for gravity force on prismatic joints at different orientations. Also, the proposed concept includes compensation of gravity for both joints in a 2-DoF case where a prismatic joint is mounted on a revolute joint. The concept requires changing the design of manipulators to include the new components that we propose. The concept depends on the analytical decoupling of effort terms and adding equivalent spring-based components to produce counterforce. A combination of those components can analytically eliminate those decoupled terms.
A-A
(b) (c)
Fig. 2. Examples of gravity compensators for prismatic joints [4]
2. Compensator for a 1-DoF prismatic joint at an arbitrary configuration
The goal of this section is to present a simple case of gravity compensation for a 1-DoF system of a prismatic joint. The aim here is to show how to compensate forces on a prismatic joint with a counterforce generated by a linear spring. Figure 3a shows a geometric representation of a prismatic joint with the moving part of mass m at an arbitrary configuration. The mass induces a reaction effort t in the actuator in the reverse direction to achieve equilibrium. Taking gravitational acceleration g pointing downwards, the actuator's effort can be as follows:
t = mg,
(2.1)
where t is the actuator's effort, m is the mass of the moving link, and g is the gravitational acceleration.
To generate counterforce, we can add a spring with stiffness k connecting the moving part of the actuator to its base. Setting the spring with a proper pre-tension s0 will generate a counterforce Fs
sp•
ks0 = Fs
s0 = F sp. (2.2)
We can achieve complete compensation by setting a proper pre-tension value s0 to generate spring force Fsp that can counterbalance the link's weight:
mg
(2.3)
As the prismatic joint performs linear motion, such a compensator construction cannot perform compensation at different joint configurations. This makes it necessary to design a mechanism that can keep this compensation force constant at any joint extension. A constant-tension mechanism is shown in Fig. 3b. A rack is coupled with the joint's slider and meshed with a pinion gear. The pinion gear is coupled with a bevel gear to transform the motion on perpendicular axes. A worm gear is coupled with the perpendicular bevel gear and meshed as input to a gear transmission to achieve motion locking. A pulley is coupled with the output of the gear transmission. The role of the pulley is to wind or unwind the wire when the slider moves up or down. A spring is fixed on the body of the joint and connected to the winding pulley through an idle pulley mounted on the moving part of the prismatic joint. This element arrangement makes the moving part of the prismatic joint supported on two parallel segments of the wire and the tension
s
0
m
(a) A linear actuator at an arbitrary configuration with spring suspension
Rack and pinion
Bevel gears
pulley
worm and worm gear
Idle pulley
(b) A geometric representation of the constant-tension mechanism
Fig. 3. Gravity compensator for a prismatic joint in vertical configuration
in this wire is generated by the spring. This makes the spring force needed half the weight of the link:
2T = mg, (2.4)
where T is the tension force in the wire generated by the spring.
From the kinematics of the system, the relationship between the joint's motion and the change in the wire's length is as follows:
Al = 2q, (2.5)
where q is the joint displacement and Al is the corresponding change in the wire's length.
This means that the retraction or expansion of the wire's length should be twice the slider's displacement q. This dictates the transmission ratio between the pinion gear and the pulley to be 1:2.
3. Compensator for a 2-DoF RP system
The goal here is to compensate gravity force for a 2-DoF RP system. Adding rotation increases the problem's complexity as the gravitational torques vary nonlinearly with rotation and the prismatic joint presents a moving center of mass. A 2-Dof RP system is shown in Fig. 4. The system consists of a revolute joint and a prismatic joint. The system consists of two masses m1 and m2 at distances lc1 and lc2 from the center of rotation, respectively. The revolute joint rotates with angle q1 and the prismatic joint slides with distance q2. This makes it possible to write the expression for the gravitational torque in the revolute joint as follows:
Ti = (lcimi + lc2m2)g cos(qi), (3.1)
where t1 is the torque effort of the revolute joint.
Fig. 4. A geometric representation of a 2-DoF RP system
As the prismatic joint slides with value q2, we can reformulate the variable lc2 as follows:
lc2 = ls0 + q2 (3-2)
where lSo defines a minimum distance between m2 and the center of rotation.
As for the effort in the prismatic joint:
r2 = m2g sin(g1),
(3.3)
where t2 is the force effort of the prismatic joint.
This equation shows that the effort in the prismatic joint is nonlinearly changing according to the rotation angle of the first joint. The pin-slot mechanism shown in Fig. 5 is designed to compensate for such nonlinearity. The mechanism consists of a slot that rotates with angle d around the point O and moves linearly through the point O. A pin p is fixed at a constant vertical distance r from point O and slides along the slot. As point p is fixed, the distance between the slot and the point changes as follows:
s = r sin(0),
(3.4)
where s is the distance between the slot and point O and r is the distance between points O and p.
O
0 = 0
(a) (b)
Fig. 5. A geometric representation of a pin-slot mechanism
It is possible to use this mechanism to compensate for the gravity effort in the prismatic joint as shown in Fig. 6. The pin-slot mechanism is used to vary the tension in the spring according
to the rotation angle q1. To statically balance the prismatic joint effort, we can properly choose spring stiffness k and pin distance r:
A^rsin^) = ^m^gsmiq^, and, accordingly, we can choose the value of spring stiffness k1 :
ki =
m2g
2 r
(3.5)
(3.6)
Rack and Bevel Pin/on
Worm and worm-gear
Pin and slot
Fig. 6. A geometric representation of a 2-DoF RP system with a prismatic joint compensator
Equations (3.5) and (3.6) show parameter selection to compensate for the gravity effort in the prismatic joint at any orientation and the mechanism presented in Fig. 3b shows how to maintain constant spring force along the prismatic joint's motion. Compensation for the revolute joint can be achieved using the construction shown in Fig. 7. We can compensate for the gravity torque in the revolute joint by connecting a spring between points A and B. Point A is fixed at a vertical distance a from the ground, while point B has an initial displacement b from the center of rotation and sliding along the link. Point B is attached to a slider mounted on a pulley-belt mechanism. The belt moves the slider with distance q* along the link, which means that the sliding of point B is a ratio of the displacement of the prismatic joint which can be achieved by gear reduction. Another spring is connected between points C and D where point C is fixed on vertical distance c and point D is fixed with distance d along the link. This allows us to write the expression for the vector representing the position of points B and D as follows:
B =
(b + q*)cos(q1) (b + q*)sin(q1)
l t
D = d cos(q1) d sin(q1)
T
(3.7)
(3.8)
Belt Pulley
Slider Pulley
Fig. 7. A geometric representation of a 2-DoF RP system with prismatic and revolute joints compensators
To choose the proper value of the spring's stiffness, we need to satisfy the equilibrium condition with a sum of torques equal to zero. The torque generated by the springs can be calculated using the cross product (B x BA) and (D x DC):
Tsp,2 = cdk2 cos(q1), (3.9)
where Tspi is the torque generated by spring CD:
Tsp,3 = a (b + q*) k3 cos(qi), (3-io)
where Tsp,2 is the torque generated by spring AB.
From (3.1), (3.9) and (3.10), we can apply the equilibrium condition to calculate the value of the springs' stiffness coefficients:
Tsp,2 + Tsp,3
(Icimi + lc2m2)g cos(qi). (3.11)
We can select our parameters to distribute the torque between both springs in a way that would eliminate the right-hand side of (3.11):
Tsp,2 = lcimig cos(qi). (3.12)
From this equation we can choose the value k2: and similarly for the second spring while substituting (3.2):
Tsp,3 = (ls0 + q^ m2g cos(qi)> (3.14)
whence the value k3 is
{k + ga)
u _ZAÏM. fQK)
For this equation to hold, the ratio between (lSo + q2 J and (b + q*) should be constant:
(^°+<?2) = HM (3 16)
(.b + q*) ak3'
This equation indicates that we can determine the distance b and the reduction ratio between q2 and q*:
b = C~¥L, (3.17)
s° ak^ 112 ak3
This means that we can control the span where point B can slide in the same way as the location of this span. This gives freedom in realizing the mechanism which can be quite complex.
4. Discussion
This system presents a concept of gravity compensation for prismatic joints for robotic systems. The concept depends on analytically decoupling joints' effort expressions and compensating them with equivalent mechanical mechanisms using linear springs. By coupling these mechanisms with the robotic system's joints, they can produce counterbalancing efforts that lead to static equilibrium without the need for the actuator's effort.
Designing a gravity compensator for prismatic joints is challenging because of the moving center of mass. Unlike links coupled with revolute joints which have a defined center of mass, prismatic joints change the location of the center of mass, which increases the nonlinearity of the actuator's effort when moving along an inclined axis. The moving spring attachment point helps tackle this problem by making the counterbalancing mechanism vary in proportion to the motion span.
The use of worm gears is important for realizing this concept. When the worm gear is of self-locking type, it can hold reverse torque through friction. This ensures the isolation of forces through the introduction of an internal reaction force that blocks any backward torque. It acts as a one-way gate to torque as it can pass the torque in one way and blocks the reverse torque. Another advantage of worm gears is that they have a high reduction ratio that reduces speed and magnifies the torque, which makes motion resistance less significant.
5. Analyzing energy and virtual work
We have introduced the synthesis method in this paper for each joint individually without taking into account the interaction forces between the compensators of both joints since the synthesis was based on analyzing the torque. However, interaction forces exist as each compensator for a corresponding joint depends on its passive adaptation on the coordinate of the other joint. This can be analyzed through the derivation of the system's energy. The energy in a spring can be expressed as follows:
Ue = ikAx2, (5.1)
where Ue is the elastic potential energy in a spring, k is the spring's stiffness and Ax is the change in the spring's free length.
From (3.5), (3.7) and (3.8) the energy in springs k1, k2 and k3 is
Uei = l-klr\iv?{ql), (5.2)
Ue2 = \k2\\Bl\\2, (5.3)
Ue 3 = (5-4)
which makes it possible to write the expression for the total elastic potential energy of the springs in the system as follows:
ue = Uei + Ue2 + XJH = \kir2 sin2 (qi) +
+ \h ((a - (b + q*) sin (qi))2 + (b + q*)2 cos2 {qS) + \k2 (d2 cos2 (qi) + (c - dsin (qi))2).
(5.5)
Substituting (b + q*) from (3.17) and taking the partial derivative to count for mechanical : gives
a tt
= (-cdk2 - gls0m2 - gm2q2 + kir2 sin (qi)) cos (qi), (5.6)
I . r. MM If/-« I ^^ til. r^.1 I ^^ t II I 1.^.1 I r-. I
(5.7)
dUe gm2 (-a2k3 sinf) + gls0m2 + gm2q2)
dq2 a2k3
The total work done by the system can be expressed as the sum of mechanical work due to gravity and mechanical work due to springs' forces. After substituting for k1, k2 and k3 from (3.6), (3.13) and (3.15) we have
dUe 1 . , , , ,
The terms in these two equations represent the effort value needed by the actuator to support the manipulator. Simulation of the system's dynamics is the method to verify the feasibility of the proposed concept.
6. Results
A joint trajectory was tested in simulation. The parameters of the manipulator and the gravity compensator were assigned according to Table 1. The parameters a, b, c, and d were assigned arbitrarily, while the values k1, k2, k3 were calculated from Eqs. (3.6), (3.13) and (3.15).
The ratios -r- and q* : q2 were calculated using Eq. (3.17). Simulation of joint torques shows
%
a complete compensation of the gravity force along the Cartesian trajectory.
The value of spring coefficients depends on their mounting location on the manipulator. Moreover, the mounting points of the springs can depend on the spring coefficient which gives more flexibility in the design process, especially, with limited space. However, the choice of
Table 1. Design parameters used in simulation
manipulator parameters compensator parameters
parameter unit value parameter unit value
k h hi h ho mm 1000 500 500 250 hn + 12 a b c d mm 400 125 400 500
m-L m2 kg 1 0.5 h k2 k3 N/rn 16.33 24.5 24.5
b % * . <7 : q 2 l 2 1:2
the mounting points of the sliding spring is limited depending on the reduction ratio between the actuator's sliding range and the spring's mounting point sliding range. This means that the mounting location for the spring compensating for the torque of the moving mass depends on the transmission ratio. It is practical to determine the mounting location prior to the transmission ratio. The transmission ratio should reduce the span of motion to keep the sliding spring within the spacial limits of the link.
— qu rad - - q2, m Trajectory
Cartesian trajectory
0.00 0.50 1.00 1.50 2.00 x, m
(a) The actuator's trajectory and the modeled effort
-un-compensated counter-torque - - compensated
7.5 5.0
a 2.5
É5 0.0 tT—2.5 -5.0 -7.5
0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0
t, s t, s
(b) The counter's effort and compensation Fig. 8. Simulation results of effort and counterbalance
Torque of first joint q1 Effort of second joint q2
Cartesian trajectory
0.0 0.2 0.4
0.8 1.0
Qi, m
Fig. 9. Modeled relationship between the prismatic joint distance q2 and the span of point B based on different transmission ratios
The relationship between the span of prismatic joint motion q2 and the span of the spring connection point B is presented in Fig. 9. We can see how the transmission ratio can affect the sliding of point B. From the graph, we can see that, if the transmission ratio is 1:1, point B needs to extend beyond the physical limits of the first link l1. However, with higher transmission values, the motion span of point B gets smaller. The value of b determines the location of point B when q2 = 0. As a design problem, either (3.15) or (3.17) can be used to determine the design
parameters. A decision of either values k3 or ^jpry can be made and then Eqs. (3.15) or (3.17) can be used to determine the other parameter.
Figure 10 shows the simulation results for compensated and uncompensated actuators' efforts considering the interaction forces between both compensators for each joint. The results show that our proposed concept of compensation significantly reduces the actuators' effort, thus reducing the energy consumed due to gravity alone.
-compensated - ■ uncompensated
Actuators effort
Actuators effort
£
7 6 5 4 3 2 1 0 -1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 q1, rad
7 6 5
£ 4 k 3 2 1 0 •1
0.00 0.05 0.10 0.15 0.20 0.25 0.30 ?2> m
Fig. 10. Comparison of compensated and uncompensated joint effort derived from system's total energy
7. Conclusion
This paper proposes a preliminary concept for a passively adapting gravity compensator dedicated to rotating prismatic joints. The compensator depends on tension springs and gear
transmission to produce the corresponding countereffort to compensate for gravity. The design of the compensator's components is parameterized to suit different constructions of manipulators and space limitations. The sliding spring mounting point provides the ability to compensate for the gravity torque generated by a sliding mass at an angle. The wire retracting mechanism provides the ability to generate constant force along the sliding range of the linear actuator.
The common robot configurations that include prismatic joints do not possess a complexity beyond what this paper has introduced. Common robot configurations with prismatic joints can be: Scara, cylindrical and spherical. Scara manipulators have one prismatic joint in a vertical configuration. Cylindrical manipulators have two prismatic joints placed perpendicularly to each other and the plane of orientation is perpendicular to gravity. The spherical manipulator has a prismatic joint mounted on a revolute joint in a configuration similar to the one in this paper. Hence, it seems enough to study a 2-dimensional case.
Conflict of interest
The authors declare that they have no conflict of interest.
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