The Puller-Follower Control of Compliant and Noncompliant Antagonistic Tendon Drives in Robotic Systems

Abstract

This paper proposes a new control strategy for noncompliant and compliant antagonistic tendon drives. It is applied to a succession of increasingly complex single-joint systems, starting with a linear and noncompliant system and ending with a revolute, nonlinearly tendon coupled and compliant system. The last configuration mimics the typical human joint structure, used as a model for certain joints of the anthropomimetic robot ECCEROBOT. The control strategy is based on a biologically inspired puller-follower concept, which distinguishes the roles of the agonist and antagonist motors. One actuator, the puller, is considered as being primarily responsible for the motion, while the follower prevents its tendon from becoming slack by maintaining its tendon force at some non-zero level. Certain movements require switching actuator roles; adaptive co-contraction is used to prevent tendons slackening, while maintaining energetic efficiency. The single-joint control strategy is then evaluated in a multi-joint system. Dealing with the gravitational and dynamic effects arising from the coupling in a multi-joint system, a robust control design has to be applied with on-line gravity compensation. Finally, an experiment corresponding to object grasping is presented to show the controller's robustness to external disturbances.

Keywords

Anthropomimetic robots antagonistic tendon drive biologically inspired control passive compliance.

1. Introduction

Anthropomimetics – copying the human body in order to construct and control similar mechanisms in the technical world – is an increasingly attractive research topic aiming to achieve human effectiveness and efficiency in those areas where engineering still cannot compete with biology (diversity of motions, manoeuvrability, etc). One of the robotic mechanisms central to anthropomimetic principles [1] is the agonist-antagonist (AA) drive. An AA drive [2] consists of two single acting actuators which act in opposition to produce a net torque at the joint: for a given motion, the actuator that initiates the acceleration in the direction of motion is termed the agonist, and the other, which may counteract the action of the agonist in order to achieve the desired quality of motion, is termed the antagonist. If the AA drive is coupled to a robotic joint using tendons and springs are then introduced into the tendons, this arrangement mimics the structure of many typical human joints.

A revolute, nonlinearly tendon coupled and compliant joint is shown in Fig. 1. (The nonlinear coupling refers to the kinematics of the coupling rather than the nature of the compliance.) It represents a simplified mechanical model of the elbow joint of ECCEROBOT (Fig. 2), an upper-body anthropomimetic robot designed as part of the European FP7 project¹ “Embodied Cognition in a Compliantly Engineered Robot”.

Figure 1.

Model of a typical human joint structure - a revolute, nonlinearly tendon coupled and compliant joint. “a” represents the brachialis muscle and “b” the triceps (the biceps muscle is omitted as it crosses two joints and is outside the analysis presented here)

Figure 2.

Real robot structure - ECCEROBOT (left) and robot model structure (right)

The idea was to create a robot as similar as possible to the human upper body, by not just replicating the appearance of a human, but by copying the inner structures and mechanisms of a human: bones, joints, muscles and tendons. Instead of using high precision actuators and stiff components as in typical humanoid robots, ECCEROBOT [3] follows the human example by using compliant materials in the drive train and in some other locations. The material used for the robot's bones is Polymorph, also known as Friendly Plastic.

It is an engineering grade caprolactone, a true thermoplastic that is easily moulded into anatomically correct shapes. A useful attribute is its distinctly bone-like appearance, which enables easy comparison with real anatomical structures, as well as adding to the robof s cosmetic appeal.

A typical joint rotation in ECCEROBOT is driven by two DC motors configured as an AA drive. The two joint motors, coupled to the bones through tendons, mimic muscles; where possible, the insertion points of these artificial muscles match those of the human. Since the robot is expected to operate in a human-centred environment and in the presence of humans, emphasis is given to achieving safe interaction with the surroundings. The safety issue is to a great extent resolved by the presence in the robot of passive mechanical compliance in the form of elastic springs in the tendons [4].

The work reported here attempts to develop a control strategy for the revolute, nonlinear and compliant joint shown in Fig. 1. For this, a new control strategy for agonist-antagonist drives using either inextensible tendons or tendons with springs has been developed and the application of the new approach to different single-joint systems has been demonstrated. The work started with a linear system with inextensible tendons and ended with a revolute, nonlinearly tendon coupled and compliant system.

In Section 2 a review of previous work that has been done relevant to the control problem we are dealing with here is presented. The principles of the new approach are stated in Section 3 and are introduced using the simplest form of the joint shown in Fig. 1, which reduces it to a linear and noncompliant system. Springs are then added to the tendons, forming a linear and compliant system which is analysed in Section 4. Finally, a fully nonlinear and compliant system is dealt with in Section 5. In Section 6 investigation is made about the extent to which the single-joint control strategy can be implemented in a multi-joint system. This entails the use of a robust control design in order to deal with the uncertain effective joint inertia and the unmodelled dynamics originating from the coupling in the multi-joint system. Gravity is then introduced into the multi-joint system and nonlinear gravity compensation, based on joint positions, is added to counteract the influence of gravity. In order to evaluate the control system performance, an experiment involving object grasping is analysed and the final results for the control of the multi-joint system are presented at the end of Section 6.

2. Biologically inspired design

The previous work forming the background for our approach includes studies of the way humans achieve the coordination of agonist and antagonist (AA) muscles [5]. Particularly interesting is the operation of AA pairs of muscles at the beginning and at the end of a movement, as well as the role exchange between agonist and antagonist during a movement. The agonist muscle initiates a movement by contracting; if the antagonist muscle contracts, it will oppose the effect of the contraction of the agonist muscle. At the beginning of the movement the nervous system sends fast stimulating (turn-on) signals to the agonist muscles and, simultaneously, reciprocal inhibitory (turn-off) signals to the antagonist muscles. The only variable that can be controlled in this process is the intensity of muscle contractions. The nervous system controls the intensity of muscle contractions in two ways: by increasing the number of motor units contracting simultaneously and by increasing the frequency of contraction of individual motor units, as is explained in the framework of the principle of summation of muscle contractions [5]. The torque generated by the controlled agonist contraction begins to turn the joint in one direction. As the end of the movement is approached, turn-off signals are sent to the agonist muscles and, simultaneously, turn-on signals are sent to the antagonist muscles. However, the changes in stimulation do not take the form of instantaneous step changes. There is a progressive activation of the larger motor units in the antagonist muscle, along with an increased frequency of stimulation of these units. At the same time, the agonist muscles turn off their motor units one by one until they reach the relaxed state. The exact sequence and speed of turning the motor units “on” and “off” is controlled by the central nervous system (CNS), and is determined by the nature of the required movement.

Numerous previous technical results constitute a useful background for the work reported here and they can be divided into several groups according to their relevance to our ideas.

2.1 Elastic joints

The first group deals with the “classical” coupling of actuators and joints. The relevant feature is the introduction of compliance into the torque transmission, which doubles the number of position coordinates. Efforts have concentrated on the modelling [6 8] and control of such systems [9 11].

2.2 Tendon driven joints

The second group considers tendon driven robotic joints, first with inextensible tendons [12 14] and later with elastic extensible types [15–16]. The main advantage of using tendon coupling compared to the classical coupling using gear-boxes is the displacement of the motor from the joint to the base of the arm, allowing for static compensation and the design of lightweight and compact manipulators. Some additional advantages of tendon-driven systems are low friction, low backlash and shock absorption during impacts. These benefits make tendons better suited than other mechanical power transmission mechanisms for applications which require small size, low weight and high-speed operation, such as artificial hands. Of course, the presence of tendons in the human body is an additional reason for selecting tendon coupling in humanoid structures.

Although tendon driven systems with inextensible tendons are of some interest to us, systems with extensible tendons are much more relevant. Elasticity in engineered tendons is usually achieved by incorporating springs into the tendons. Introducing compliance into torque transmission increases the safety of the robotic manipulator, but controlling such manipulators is often complex and troublesome, and is seen as a major disadvantage. Lee and Lee [15] and Chang, Lee and Yen [16] developed a dynamic analysis of tendon-driven manipulators with flexible tendons, and a control system design is discussed in the papers included in the group below.

2.3 Revolute, compliant, tendon drivenjoints with drive redundancy

The third group consists of the papers most relevant to the problem of controlling revolute compliant robot joints with drive redundancy.

Zinn, Roth, Khatib and Salisbury proposed a new actuation approach (DM²) for human-friendly manipulator design [17]. The main characteristics of this approach are tendon transmission, passive compliance and drive redundancy. This approach partitions torque generation into low- and high-frequency components and assigns these components to the high power, main drive and the low power, supporting drive, respectively. The effectiveness of this distribution can be seen clearly having in mind that most manipulation tasks involve position or force control, which are dominated by low-frequency torques [17]. High-frequency torques are almost exclusively used for disturbance rejection. Additionally, the DM² approach distributes the torque generation between the main and the supporting drive, according to the torque magnitude. The main drive is responsible for generating high torques, thus the safety of the manipulator primarily depends on the coupling of this motor with the manipulator (since the supporting motor can generate only low torques). The main drive is coupled to the manipulator following the series elastic actuator approach [8], which places a spring between the motor and the manipulator.

Another biologically inspired, lightweight and elastic “bionic” robot manipulator was designed within the BioRob project [18]. The drive structure used was very similar to the DM² approach, but the method used to achieve compliance was different, in that the DM² approach introduced a spring at the output of the main motor, while the “bionic” manipulator achieved compliance by placing springs in the tendons.

Although the mechanisms mentioned above include drive redundancy, this differs from the drive redundancy of agonist-antagonist drives. In contrast to the torque distribution according to torque magnitude used in the above-mentioned papers, agonist-antagonist drives employ torque distribution by motor function. In the DM² and “bionic” approaches, the motors operate in parallel, and thus the generated moments are summed. In the agonist-antagonist approach, the motors work on opposite sides of the joint, one opposing the other, and thus the moments are subtracted.

Jacobsen, Ko, Iversen and Davis proposed several control strategies for an electrical AA drive applied to a revolute joint [19], although the original idea behind these control schemes seems to have come from the “UTAH/MIT Dextrous hand” [20]. The control strategy called “position and force control with rectifiers” [17], shown in Fig. 3, involves measuring the joint position and the forces in the tendons (F_af and F_ae). The main controller is a position controller, implemented with proportional and differential compensation. The output of this controller, the “compensated” position error, is analysed using flexion and extension rectifiers. The outputs of the rectifiers are positive and are added to the co-contraction value, thus forming the references for the tension forces.

Figure 3.

Position and force control with rectifiers applied to the agonist-antagonist drive

Force references created in this way are always positive, ensuring that the tendons will always be in tension (slack tendons introduce backlash in the torque transmission, thus being undesirable). Force compensators are included to help the actuators to accurately maintain force-bias commands. The function of the force compensator is:

T (s) = K (τ_{1} s + 1) / [(τ_{2} s + 1) (τ_{3} s + 1)]

(1)

It behaves similarly to a derivative and integral controller, depending on the values of τ₁, τ₂, and τ₃. Furthermore, high-frequency components are filtered by the pole with time constant τ₃. It will be shown that this control scheme appears to be adequate for the noncompliant and linear robotic system which is considered in Section 3, but it is not appropriate for a compliant system, discussed in Section 4, or for a nonlinear system, which is discussed in Section 5.

However, previous research has not paid any attention to “switching” – the process of exchanging roles between the two opposed actuators – but instead has assumed that both tendons are always in tension. This neglect leads to an energetically inefficient control, because the level of co-contraction is significant in relation to the net torque. Our proposal is to consider one actuator, the puller, as being responsible for the joint motion, while the role of the other is to maintain some minimum tension in the inactive tendon. Moreover, an adaptive co-contraction value is introduced, which lowers the energy consumption arising from opposing forces in co-contraction. This arrangement can be seen as more human-like in some ways, however, it requires the effective and efficient control of switching.

3. The principles of the ‘new approach'

The principles of the new approach will be explained using a simplified, linear and noncompliant single-joint system, shown in Fig. 4. This system consists of an electrical AA drive linearly coupled to the object (the robot's limb segment) using inextensible tendons. In this horizontal experiment, gravity does not influence the segment's motion. The dynamics of DC motors and the equations of gear-boxes and reels apply to both the “a” and “b” sides in the same form, and therefore the superscripts a and b will be omitted. Thus, equations (2,3) stand for the DC motors, equations (4,5) for the gearboxes and equations (6,7) for the reel torques:

C^{M} i = I^{r o t} \ddot{θ} + B \dot{θ} + M

(2)

u = R i + C^{E} \dot{θ}

(3)

θ^{r} = \frac{θ}{N}

(4)

M^{r} = μ N M

(5)

M^{r} = F r (for the “a” side)

(6)

M^{r} = - F r (for the “b” side)

(7)

Figure 4.

Agonist-antagonist drive applied to a single-joint, linear and noncompliant system

where C^M is the constant of torque, I^rot is the rotor moment of inertia, B is the viscous friction coefficient, M is the motor output torque, u is the input voltage, R is the armature resistance, i is the motor current, C^E is the coefficient of back electro-motive force, θ is the angle of motor shaft rotation, N is the gear-box ratio, μ is the gear-box efficiency coefficient, r is the reel radius, θ^r is the reel angle (gear-box output angle), M^r is the gear-box output torque and F is the tension force in the tendon (tendon force).

The equation of motion of the object is:

m \ddot{q} = F^{a} - F^{b}

(8)

Since the motors and the segment are coupled and it is an inextensible connection, the coordinates q, θ^a and θ^b are not independent. The following relations hold:

θ^{a} = \frac{N^{a}}{r^{a}} q

(9)

θ^{b} = \frac{N^{b}}{r^{b}} q

(10)

Therefore, the system has only one degree of freedom (DOF). Relations (1–10) form the complete set of equations that describe the system in Fig. 4. From these equations the following dynamic model of the system is obtained:

\ddot{q} = - \frac{s}{r} \dot{q} + \frac{c^{a}}{r} u^{a} + \frac{c^{b}}{r} u^{b}

(11)

Where the constants r, s, c^a, and c^b are:

s = \frac{μ^{a} N^{a^{2}}}{r^{a^{2}}} (B^{a} + \frac{C^{M a} C^{E a}}{R^{a}}) + \frac{μ^{b} N^{b^{2}}}{r^{b^{2}}} (B^{b} + \frac{C^{M b} C^{E b}}{R^{b}})

(12)

r = m + \frac{μ^{a} N^{a^{2}} I^{r o t a}}{r^{a^{2}}} + \frac{μ^{b} N^{b^{2}} I^{r o t b}}{r^{b^{2}}}

(13)

c^{a} = \frac{μ^{a} N^{a} C^{M a}}{r^{a} R^{a}}

(14)

c^{b} = \frac{μ^{b} N^{b} C^{M b}}{r^{b} R^{b}}

(15)

The equations of the tension forces F^a and F^b are:

F^{a} = F_{C 1}^{a} \dot{q} + F_{C 2}^{a} u^{a} + F_{C 3}^{a} u^{b}

(16)

F^{b} = F_{C 1}^{b} \dot{q} + F_{C 2}^{b} u^{a} + F_{C 3}^{b} u^{b}

(17)

Where the constants r, s, c^a, and c^b are:

F_{C 1}^{a} = \frac{μ^{a} N^{a^{2}}}{r^{a^{2}}} [\frac{I^{r o t a} s}{r} - (B^{a} + \frac{C^{M a} C^{E a}}{R^{a}})]

(18)

F_{C 2}^{a} = c^{a} (1 - \frac{μ^{a} N^{a^{2}} I^{r o t a}}{r^{a^{2}} r})

(19)

F_{C 3}^{a} = - \frac{μ^{a} N^{a^{2}} I^{r o t a} c^{b}}{r^{a^{2}} r}

(20)

F_{C 1}^{b} = \frac{μ^{b} N^{b^{2}}}{r^{b^{2}}} [- \frac{I^{r o t b} s}{r} + (B^{b} + \frac{C^{M b} C^{E b}}{R^{b}})]

(21)

F_{C 2}^{b} = \frac{μ^{b} N^{b^{2}} I^{r o t b} c^{a}}{r^{b^{2}} r}

(22)

F_{C 3}^{b} = c^{b} (- 1 + \frac{μ^{b} N^{b^{2}} I^{r o t b}}{r^{b^{2}} r})

(23)

When compared to the original joint (Fig. 1), this simplified system possesses two important characteristics: tendon coupling and drive redundancy.

The main feature of tendon coupling is that of unidirectional power transmission, i.e. tendons can pull, but not push. Because of this inability to transfer compression, the tendon can slacken when its tension force becomes zero, leading to undesirable backlash in the torque transmission. At this point, the segment coordinate and the motor angle will become independent variables. This presents a problem because the dynamic model of (11) would then not hold. This potential difficulty can be eliminated by preventing the tendon from becoming slack. In other words, it is henceforth assumed that the tendon will always be under tension and that the model (11) is always valid. The control system now has an additional task – maintaining some appropriate tension in the tendon.

Let us now consider the problem of drive redundancy. Let us assume that the segment should be moved in a positive direction, i.e. to the right. With the resting state as the initial condition, this means that the tendon forces “a” and “b” are equal. Moving the object can be achieved by following one of two possible scenarios: by increasing force F^a(changing the voltage of motor “a”), or by decreasing force F^b (changing the voltage of motor “b”). The second scenario does not meet our requirements since it carries the risk of a tendon slackening, so the first scenario is preferred. According to this scenario:

•

the tendons are always in tension (or stretched) and so both motors are active;

•

the pre-tension (tonus) is relatively small;

•

movement of the segment is achieved by increasing the voltage of one motor, the “puller”.

It is now possible to define the main control requirements for an isolated joint. The basic control requirement is the control of the joint position. As noted earlier, there is an additional control task of maintaining the appropriate tensions in the tendons. By “appropriate tension”, it is assumed a tension above some minimal value that will guarantee that the tendons will remain stretched during joint motion. This minimum will be called the reference tension force. The reference force will be defined as an input parameter in the control system and will be determined by the designer of the control system. Accordingly, for each joint there are two references: the reference position (i.e. the reference trajectory) and the reference tension force.

Now, defining the control strategy for the electrical AA drive is possible. The main idea of the approach lies in the separation of roles between motors. The two possible roles are:

the pulling role – moving the segment;

the following role – keeping the tendon under appropriate tension.

Depending on the segment motion, one of the motors will be the puller, while the other will be the follower. When a requirement for a different motion arises, the need for exchanging roles (“switching”) may occur. The motor that was the puller will then become the follower, and the follower will take over the pulling role. It is important to note that the motors always have opposing roles. As an example of a change of motion, consider a triangular velocity profile. It is required that the segment should accelerate during the first half of the motion interval, and decelerate in the second half. In order to achieve the acceleration, it is necessary for motor “a” to initiate the motion – motor “a” is therefore the puller, and motor “b” is the follower. When, at the midpoint of the motion interval, the need for deceleration arises, motor “b” will take over the pulling role while motor “a” will become the follower.

Switching occurs when the actual pulling force in tendon “a” decreases below the reference tension force. After switching, the reference force and force control are applied to the “new follower”.

In Section 4 it will be demonstrated how it is possible to use this control strategy in order to control an AA drive applied to linear joints, with either inextensible tendons or tendons incorporating springs, and Section 5 will show how one could control an AA drive applied to a joint which mimics the typical human joint – a revolute, nonlinearly tendon coupled, and compliant joint.

4. Control synthesis of linear joints with AA drive

This Section presents our control approach to the robot joint with antagonistically coupled drives, in the very beginning without compliance (Fig. 4), and later including compliance through the linear elastic springs (Subsection 4.2). In this way, presented work gradually moved towards the target structure of the anthropomimetic revolute, nonlinear, and compliant robotic joint.

4.1 Noncompliant case

In this Section the linear and noncompliant system shown in Fig. 4 is considered. As discussed in Section 3, there are two control tasks: controlling the motion of the segment, and controlling the force in the following tendon. A PID regulator will be employed for each of the tasks and for each motor. This means that our control system will include four PID regulators. At any time this two-input two-output (TITO) system will be controlled by using two single-input single-output (SISO) controllers. The trajectory and the force tracking performance obtained from Matlab simulations are shown in Fig. 5.

Figure 5.

Trajectory tracking and force tracking in the linear and noncompliant system with AA drive

It can be seen that the trajectory tracking performance had a tracking error of less than 2% and was not compromised by switching. However, the switching delivered a shock to the force controller and oscillations appeared after the switching instant. The transient period was only about 12ms, but the magnitudes of the oscillations were unacceptably large, leading to negative values for the tendon forces. The appearance of the shock in the force tracking can be explained by the fact that the tension forces, equations (16, 17), are functions not only of the state variables, but also of the state derivatives, and thus can change discontinuously.

Therefore, the control system did not achieve the target of maintaining the reference tension at all times. The solution to this problem can be found by examining the effects on the control system of changing the reference tension. Our Matlab simulations revealed that the switching shock strongly depends on the level of the reference tension force, as shown by the dotted line in Fig. 6. It can be seen that the minimum shock appeared when the reference tension force was about 25% of the actual pulling force. The minimum values of the tension forces during switching (shown by the dashed line in Fig. 6) are also examined. It can be seen that if the reference tension force from 19% to 40% of the actual pulling force is set, then both tension forces will be positive (i.e. both tendons will be taut) all the time. Here the idea of the maximal tension margin is introduced, which is the value of the reference tension force (as a percentage of the actual pulling force) at which the minimum value of the tension forces during switching is highest. This occurs when the reference tension force is close to 32% of the pulling force. Therefore, one can choose the reference tension in order to gain either of two advantages: the minimal switching shock (with the reference tension set to 25% of the actual pulling force) or the maximal tension margin (with the reference tension set to 32% of the actual pulling force).

Figure 6.

The trade-off between the minimal switching shock and the maximal stretching margin

In order to avoid unnecessary energy consumption, we can take advantage of the fact that this higher reference force need not be applied all the time. It can be increased a short time before the switching takes place and reduced again shortly after the switching, thus becoming an “adaptive reference force”. This means that, at the time of switching, the reference could have some optimal value for both reducing the shock to some acceptable level and ensuring a suitable tension margin. The duration of this high-force interval need not be long, and thus the increased force need not affect greatly the energy consumption. A new idea of fuzzy adaptive reference force activity has been developed. It proposes activation of the tension reference increase under the next conditions (according to the human elbow muscles activity [21,22]):

•

the puller force is decreasing with the slope steeper than 70°;

•

the value from which puller force function starts rapidly to decline is higher than 200% of tension reference force value.

The simulations of changing the reference tension revealed another effect. Sudden changes (i.e. step changes) in the reference tension force deliver a separate shock to the force controller, generating oscillations. To prevent this new shock, the reference force must change smoothly.

The final results for controlling the electrical AA drive applied to the linear and noncompliant joint system are shown in Fig. 7. Trajectory tracking is still adequate for many potential applications (with a tracking error of less than 2%); the switching shock still exists but it is reduced to an acceptable level, and the tendons are taut all the time (the tension forces are always above 0.2 N). Since the same parameters for both motors were adopted, it was therefore possible to use the same position controllers (24) and likewise for the force controllers (25):

K_{p}^{a} (s) = K_{p}^{b} (s) = \frac{33 s + 5000}{0.0006 s + 1}

(24)

K_{f}^{a} (s) = K_{f}^{b} (s) = \frac{0.014 s + 80}{s}

(25)

Figure 7.

Final trajectory tracking and force tracking in linear and noncompliant system with AA drive

4.2 Compliant case

In this Section a linear and compliant system shown in Fig. 8 is considered. The compliance is introduced by adding springs in series with the tendons. This system is very close to the final system – it features tendon coupling, drive redundancy, and compliance, except that the tendon coupling is linear, instead of the nonlinear coupling of the final system. It has 3 DOFs, and accordingly 3 independent position coordinates and 6 state variables. This system is completely controllable. However, the fact that the system now has passive compliance does not in itself ensure that the tendons will not slacken during the segment motion. Therefore, the ideas introduced in Section 3 of different motors having different roles, and of the switching of those roles, are also useful here.

Figure 8.

Agonist-antagonist drive applied to a linear and compliant system

To control the system, a multivariable feedback approach based on decoupling is applied – a two-step compensator design approach [23]. This approach involves, first, the design of a pre-compensator to deal with the interactions within the system, and second, the design of a diagonal controller using SISO theory. The plan is that the pre-compensator will transform the original compliant system into a state in which the idea of the separation of roles between the motors can be applied.

The diagonal controller, in the form of one position and one force controller, is very similar to the controllers (24) and (25) designed in Section 4.1. Moreover, the adaptive reference tension force, which increases its value during the switching transition, as described in Section 4.1, is also used here. The trajectory tracking and the force tracking performances are shown in Fig. 9.

Figure 9.

Trajectory tracking and force tracking in the linear and compliant system with AA drive

The trajectory tracking is again satisfactory for many potential applications and is not compromised by switching. The force tracking is also adequate in the steady state periods before and after switching. Most importantly, there are no oscillations in the transient phase after switching and this is clearly the major positive effect of adding compliance. This could be explained by the fact that the tension forces are functions of the state variables alone and thus cannot change discontinuously. An undesirable negative overshoot appeared in the direction of decreasing force, but this was completely eliminated by applying the adaptive reference tension force. The final conclusion is that both the trajectory tracking and the force tracking in the linear and compliant system are sufficiently accurate and stable for many potential applications.

5. Control of a revolute, nonlinearly tendon coupled and compliant joint – the human-like joint structure

This section elaborates upon the original system shown in Fig. 1. Before the start of the system dynamics discussion, a brief explanation of the symbols used in Fig. 1 is given and some geometrical relations that exist in the system are identified. The geometrical parameters are: l_a1 (l_b1) – the distance between the joint axis and reel “a” (reel “b”); l_a2 (l_b2) – the distance between the joint axis and the point where tendon “a” (tendon “b”) is connected to the lower limb segment. q denotes the joint angle and its zero position corresponds to the “extended” joint. Symbols ξ^a and ξ^b denote the actual length of tendon “a” (i.e. the length of cable “a” plus the length of spring “a”) and the actual length of tendon “b”, respectively, and can be determined as follows (neglecting the reel radius):

ξ^{a} (q) = \sqrt{l_{a 1}^{2} + l_{a 2}^{2} + 2 l_{a 1} l_{a 2} cos (q)}

(26)

ξ^{b} (q) = \sqrt{l_{b 1}^{2} + l_{b 2}^{2} - 2 l_{b 1} l_{b 2} cos (q)}

(27)

Angles ϕ^a and ϕ^b are the angles between the lower limb segment and tendons “a” and “b”, respectively, and are determined as follows:

ϕ^{a} = arcsin (\frac{l_{a 1} sin (q)}{ξ^{a} (q)})

(28)

ϕ^{b} = arcsin (\frac{l_{b 1} sin (q)}{ξ^{b} (q)})

(29)

The dynamic of the lower limb segment motion is:

I \ddot{q} = l_{a 2} sin (ϕ^{a}) F^{a} - l_{b 2} sin (ϕ^{b}) F^{b}

(30)

where I is the lower limb segment moment of inertia. The elastic (tension) forces are assumed linear² and consist of two components: one proportional to elastic deformation and the other differential (damping), as shown in (31, 32), where k^a and k^b are spring stiffness constants and d^a are d^b are damping constants. Δl^a and Δl^b are deformations (tendon extensions) equal to (33, 34), where is the initial length of tendon “a” (“b”). Linear spring characteristics prevent any kind of stiffness control [24].

F^{a} = k^{a} Δ l^{a} + d^{a} Δ {\dot{l}}^{a}

(31)

F^{b} = k^{b} Δ l^{b} + d^{b} \dot{Δ l^{b}}

(32)

Δ l^{a} = ξ^{a} (q) - ξ_{0}^{a} + r^{a} θ^{r a}

(33)

Δ l^{b} = ξ^{b} (q) - ξ_{0}^{b} - r^{b} θ^{r b}

(34)

The system has three DOFs: coordinates q, θ^a and θ^b. The state vector consists of the following: the angle of motor “a” (x₁) and its angular velocity (x₂), the angle of motor “b” (x₃) and its angular velocity (x₄), the joint angle (x₅) and the joint angular velocity (x₆). The equations of the tension forces are:

\begin{array}{l} F^{a} & = k^{a} (ξ (x_{5}) + \frac{r^{a}}{N^{a}} x_{1} - ξ_{0}^{a}) \\ + d^{a} (- \frac{l_{a 1} l_{a 2} \sin (x_{s})}{ξ (x_{s})} x_{6} + \frac{r^{a}}{N^{a}} x_{2}) \end{array}

(35)

F^{b} = k^{b} (ξ^{b} (x_{5}) - \frac{r^{b}}{N^{b}} x_{3} - ξ_{0}^{b}) + d^{b} (\frac{l_{b 1} l_{b 2} sin (x_{5})}{ξ^{b} (x_{5})} x_{6} - \frac{r^{b}}{N^{b}} x_{4})

(36)

Comparing this system to the linear, compliant system discussed in Section 4.2, where the tension forces were linear combinations of the state variables, here the tension forces are nonlinear combinations of state variables. The state space model of the revolute, nonlinearly tendon coupled and compliant system is:

\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{2 p} (x_{1}, x_{2}, x_{5}, x_{6}) + \frac{C^{M a}}{R^{a} I^{r o t a}} u^{a} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = x_{4 p} (x_{3}, x_{4}, x_{5}, x_{6}) + \frac{C^{M b}}{R^{b} I^{r o t b}} u^{b} \\ {\dot{x}}_{5} = x_{6} \\ {\dot{x}}_{6} = x_{6 p} (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) \end{matrix}

(37)

where x_2p, x_4p, and x_6p are nonlinear functions of the state variables:

\begin{array}{c} x_{2 p} (x_{1}, x_{2}, x_{5}, x_{6}) = - \frac{r^{a^{2}} k^{a}}{μ^{a} N^{a^{2}} I^{r o t a}} x_{1} \\ - \frac{1}{I^{r o t  a}} (\frac{r^{a^{2}} d^{a}}{μ^{a} N^{a^{2}}} + (B^{a} + \frac{C^{M a} C^{E a}}{R^{a}})) x_{2} \\         + \frac{r^{a}}{μ^{a} N^{a} I^{r o t  a}} (- k^{a} ξ^{a} (x_{5}) + \frac{d^{a} l_{a 1} l_{a 2} sin (x_{5})}{ξ^{a} (x_{5})} x_{6} + k^{a} ξ_{0}^{a}) \end{array}

(38)

\begin{array}{c} x_{4 p} (x_{3}, x_{4}, x_{5}, x_{6}) = - \frac{r^{b^{2}} k^{b}}{μ^{b} N^{b^{2}} I^{r o t b}} x_{3} \\ - \frac{1}{I^{r o t b}} (\frac{r^{b^{2}} d^{b}}{μ^{b} N^{b^{2}}} + (B^{b} + \frac{C^{M b} C^{E b}}{R^{b}})) x_{4} \\ + \frac{r^{b}}{μ^{b} N^{b} I^{r o t b}} (k^{b} ξ^{b} (x_{5}) + \frac{d^{b} l_{b 1} l_{b 2} sin (x_{5})}{ξ^{b} (x_{5})} x_{6} - k^{b} ξ_{0}^{b}) \end{array}

(39)

\begin{array}{c} x_{6 p} (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) = \frac{λ_{a} (x_{5}) r^{a} k^{a}}{N^{a} I} x_{1} + \frac{λ_{a} (x_{5}) r^{a} d^{a}}{N^{a} I} x_{2} \\ + \frac{λ_{b} (x_{5}) r^{b} k^{b}}{N^{b} I} x_{3} + \frac{λ_{b} (x_{5}) r^{b} d^{b}}{N^{b} I} x_{4} + \frac{1}{I} λ_{a} (x_{5}) k^{a} ξ^{a} (x_{5}) \\ - \frac{1}{I} λ_{b} (x_{5}) k^{b} ξ^{b} (x_{5}) - \frac{1}{I} λ_{a} (x_{5}) k^{a} ξ_{0}^{a} + \frac{1}{I} λ_{b} (x_{5}) k^{b} ξ_{0}^{b} \\ - \frac{1}{I} (\frac{λ_{a} (x_{5}) d^{a} l_{a 1} l_{a 2}}{ξ^{a} (x_{5})} + \frac{λ_{b} (x_{5}) d^{b} l_{b 1} l_{b 2}}{ξ^{b} (x_{5})}) s i n x_{5} x_{6} \end{array}

(40)

Where I is the moment of inertia of the lower limb segment, and λ_a(x₅) and λ_b(x₅) are functions introduced to simplify the above expression and are defined as:

λ_{a} (x_{5}) = \frac{l_{a 1} l_{a 2} sin (x_{5})}{ξ^{a} (x_{5})}

(41)

λ_{b} (x_{5}) = \frac{l_{b 1} l_{b 2} sin (x_{5})}{ξ^{b} (x_{5})}

(42)

The idea for designing the control system for this type of joint follows from Section 4.2, in which the linear and compliant systems are discussed. The only difference is that now nonlinearity is included. In Section 4.2, multivariable feedback control based on decoupling was found to be successful. Therefore, a similar logic is followed for the control strategy here. Firstly, both decoupling and linearization by applying input-output feedback linearization for MIMO system [25] is achieved (feedback linearization also appeared as the best solution for a robotic joint with classical drive including stiffness [26]). After that, SISO controllers can be designed for the new system, since the feedback linearization will have shaped the original compliant system into a form in which the idea of separating the roles of the motors can be applied. The SISO controllers are very similar to controllers (24) and (25), designed in Section 4.1. Moreover, the adaptive reference tension force can also be used, as described in Section 4.1.

The complete control scheme for a revolute, nonlinearly tendon coupled and compliant joint with AA drive is shown in Fig. 10.

Figure 10.

The control scheme for the AA drive applied to the revolute, nonlinearly tendon coupled and compliant joint

The results of Matlab simulations of the trajectory tracking and the force tracking are shown in Fig. 11. Our final conclusion is that the trajectory tracking and the force tracking in the revolute, nonlinear and compliant joint are fully acceptable for many potential applications.

Figure 11.

Trajectory tracking and force tracking in revolute, nonlinear and compliant robotic joint

6. Robustness to coupling in a multi-joint system

In this Section examination will be made of the extent the single-joint control strategy developed above can be implemented in a multi-joint system through the ECCEROBOT hardware architecture [27]. The full multi-joint dynamic model of the ECCEROBOT as a humanoid robot with compliant antagonistically coupled drives is presented in reference [28]. Contact analysis as a key feature of the robot aimed to work in a human-centred environment is demonstrated in reference [29]. As far as joint control is concerned, the new features are that the joint inertia and gravity loads now depend on the positions of other joints and that dynamic coupling exists between joints. Initial simulation experiments showed that the controllers developed for single joints could not cope with the new situation – the nonlinear parts of the control system of Section 5 could not completely counteract the new effects and so the control performance was compromised.

The solution for this problem proposed in this paper is a robust control system design, together with nonlinear gravity compensation.

6.1 Gravity compensation

The state space model of the revolute, nonlinearly tendon coupled and compliant joint (37) did not consider gravity. The aim of this section is to include gravity in the model and to define the necessary gravity compensation.

Each joint in a multi-joint system is influenced by the static gravitational moment (M_g) which can be easily calculated when the whole system position is known (from the robot dynamic model reduced to static). Introducing gravity into the state space model (37) results only in a change in eq. (40) as follows:

{\dot{x}}_{6 N E W} = x_{6 p N E W} (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) = x_{6 p} - \frac{1}{I} M_{g}

(43)

This will change the corresponding values in the decoupling matrix in the feedback linearization, thus making the control system aware of the influence of gravity.

6.2 Uncertain effective joint inertia and dynamic coupling between joints

This section details how to deal with the two remaining effects that can appear in joints of a multi-joint robot body: changeable joint inertia and the consequences of dynamic coupling between joints.

In contrast to the single-joint system shown in Fig. 1, where the moment of inertia of the moving segment was constant and known, in a multi-joint system the effective joint inertia changes and in general it depends on the current position of the entire system. From the single-joint viewpoint, the effective inertia therefore constitutes an uncertain parameter. On the other hand, the coupling between joints presents itself as unmodelled dynamics. In order to deal with both the uncertainty and the unmodelled dynamics, a robust control system will be designed. Because of the applied input-output feedback linearization, the uncertainty in the system cannot be described as a function of the system's parameters; therefore the H_∞ loop-shaping method proposed by McFarlane and Glover [30] will be used - a combination of loop shaping and robust stabilization. Instead of the four existing SISO controllers (the position controllers K_{p I} and K_{p II} and the force controllers K_{f I} and K_{f II}, shown in Fig. 10) four new SISO controllers are proposed. Since the same parameters for both motors were adopted, it was possible to use the same position K_p and force K_f controllers:

K_{p}^{I} (s) = K_{p}^{II} (s) = \frac{9.15 e 9 s^{3} + 2.6 e 11 s^{2} + 4.2 e 12 s + 2.8 e 13}{s^{3} + 6.3 e 3 s^{2} + 3.9 e 5 s + 8.9 e 7}

(44)

K_{f}^{I} (s) = K_{f}^{I I} (s) = \frac{1.9 e 4 s + 7.6 e 5}{s + 2.7 e 2}

(45)

Until now, a robust control system with gravity compensation for a particular joint has been proposed. In a simulation experiment, this single-joint control strategy will be used to control two joints in a multi-joint system. In order to test the robustness of the control system, and in addition to the existing uncertainty and unmodelled dynamics, some external disturbances will be introduced. The experiment consists of raising the arm in the sagittal plane while moving the elbow from 15° to 45° and the shoulder from 0° to 45°, in 2 seconds, following a triangular velocity reference (uniform acceleration followed by uniform deceleration.

During this movement, at t=1s, the robot grasps a 0.5 kg ball and holds it until the end of the task – this is regarded as a strong external disturbance. The trajectory tracking and the force tracking in the elbow joint and the shoulder joint (joints 10 and 9 respectively in Fig. 2b) during this grasping experiment are shown in Fig. 12. The control system shows very acceptable behaviour - the influence of the external disturbance is reduced after 1s to less than 2% position error, steady state position errors do not exist, and the tendons are taut at all times during the movement, even though the reference tension force was set to the very low value of 3 N (which makes the system energetically very efficient).

Figure 12.

The trajectory and the force tracking during the grasping experiment in the multi-joint system

7. Conclusion

The paper dealt with the complex problem of modelling and controlling a human-like joint - a revolute, nonlinearly tendon coupled and compliant joint. The complexity of the system originated mainly from its drive system, which involved two antagonistically working motors per joint, coupled to the joint by tendons with passive compliance. The drive system made the motor motions independent of the joint motions, thus tripling the total number of DOFs. This redundancy was successfully resolved by proposing the puller-follower concept, which at any time separates the roles of the two joint motors, one being the puller responsible for the joint motion and the other being the follower that keeps the inactive tendon from slackening. An adaptive force reference (adaptive co-contraction) was needed to properly control the effects of role switching between motors and also to provide an energetically efficient control algorithm.

This new approach for controlling AA drives was first illustrated using a linear and noncompliant robotic joint and was then extended to a linear and compliant joint using multivariable control theory. Finally, it was dealt with using a revolute, nonlinearly tendon coupled and compliant joint, where a multivariable feedback approach

–

input-output feedback linearization for MIMO systems

–

was successfully applied. The single-joint control strategy was then evaluated at a joint of a multi-joint system. With a multi-joint robot body, the effective joint inertia and the gravitational load both depend on the position of the entire system; in addition, dynamic coupling between joints was present, strongly disturbing the control of the joint. Robust control theory was successfully applied to compensate for the disturbances and enable acceptable control of the entire system.

The conclusion is that the proposed puller-follower concept, employing a human-like separation of roles between the agonist and antagonist actuators, combined with the use of advanced control theory, can lead to the successful control of agonist-antagonist drives. Our experiments have demonstrated that high-quality trajectory tracking, successful control of tendon force, efficient switching and energy efficiency can be achieved.

Footnotes

8.

The research leading to these results has received funding from the European Community's Seventh Framework Programme FP7/2007–2013 - Challenge 2- Cognitive Systems, Interaction, Robotics - under grant agreement no. 231864- ECCEROBOT; and partly by the Serbian Ministry of Science and Technological Development under contracts 35003 and 44008.

1

www.eccerobot.org

2

Measurements on springs used in the ECCEROBOT are done by the project partner: AI lab, University of Zurich

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