The torque generated by a motor is transmitted through a flexible cable in a cable-driven system. The force or torque in two different directions is required to drive a rotational joint, but the cable cannot transmit forces in the pushing direction. Three different basic cable-driven mechanisms have been used to satisfy these requirements, as shown in Fig. 1.

As shown in Fig. 1a, the mechanism can drive one joint with one motor. The structure is simple, so it is widely used in many robotic manipulators. However, sometimes it is difficult to ensure that the non-pulled side is tensioned, which will cause backlash and slow responses. This will seriously affect the performance of the mechanism, so it is essential to add a pre-tensioning mechanism, which will add to the complexity of the system.

In the second mechanism, shown in Fig. 1b, each cable is independently driven by one motor. This mechanism provides high performance and it is easy to design the cable routing structure. The joint stiffness can be easily adjusted by controlling the two motors. However, the cost is higher, as the number of required motors is doubled compared with other mechanisms, and application is limited by the requirement of compact space.

As is shown in Fig. 1c, the third mechanism uses a spring for the reversed actuation of one side, and a motor is used to drive the other side. The cables can be tensioned by pulling the spring to an appropriate distance. This will apply two directional forces to the joint. The cable tension can always be maintained without an additional pre-tensioning mechanism. Due to simple cable routing, the appropriate cost and the character of the tensioning itself makes the spring-motor mechanism a good choice when compact structures are required.

From the analysis above, the spring-motor mechanism (shown in Fig. 2c) was selected for the design of the micromanipulator.

The mechanism of the micromanipulator for a surgical robot is shown in Fig. 2a. The four different colours represent the drive mechanism of different joints, with each joint designed based on the spring-motor mechanism. The micromanipulator has four DOFs, and the wrist of the micromanipulator has 3 DOFs, consisting of one pitch joint and two deflection joints. The driving force of each joint is determined by the difference between the tensions of the cables in each loop of the joint cable routing. The opening, closing, and deflection of the wrist are determined by the different combinations of the rotation directions of the motors used in gripper A and gripper B. The 3-D model of the wrist is shown in Fig. 2b. The wrist and the driven components are driven by a rotary joint. The components of each joint are similar to those of the single joint prototype, as shown in Fig. 3.

The dynamic equation of operating forceps is shown in Eq. (1) and takes the external force of the system and the joint friction torque into consideration. τi=Di(q)q¨+Hi(q,q˙)+Gi(q)+τfi+JiTFexti. Where i=a,b, a and b represent gripper A and gripper B respectively. τi, Di (q), Hi (q,q˙), Gi (q), τfi, JiT and Fexti are the joint driven torque, inertia matrix, centrifugal force and Coriolis force, gravity, friction torque, force Jacobian matrix and external force respectively. The driving torque of each joint is shown in Eq. (2); the meaning of the superscript i is similar to Eq. (1). Where, FM is the tension of the positive driven cable. FS is the tension of the negative driven cable. τ1=r1FMr-FSrτ2=r2FM-FSτ3i=r3FMi-FSiτi=τ1τ2τ3iT. When the external force is zero, the micromanipulator moves freely. The dynamic equation of the operating forceps is shown in Eq. (3), and the parameters can be obtained by inverse dynamics identification. τ^i=τ^1τ^2τ^3iT=D^i(q)q¨+H^i(q,q˙)+G^a(q)+τ^fi. When the external force is not equal to zero, the external force can be obtained from Eq. (4), where τi is the actual joint torque and τ^i is computed by the parameters in Eq. (3). Digital filtering can be used first to obtain the data of τi and τ^i. Fexti=Ji-Tτi-τ^i. The external force Fext applied on the micromanipulator and the clamping force Fcp of the micromanipulator can be described as in Eq. (5). Fext=Fexta+FextbFcp=12Fextya+Fextyb.

The four joints drive the micro devices and the cable routing structure as presented in the analysis above. The performance of the single joint principle prototype can illustrate the performance of the micromanipulator. The prototype of the single joint model was designed as shown in Fig. 3. The DC servo motor provides the positive driving force and the return spring provides the negative driving force. The displacements of the flexible cables were measured by two sets of raster detection modules and a non-contact micro magnetic encoder was used to detect the joint angle of the gripper. A photoelectric switch provides the origin signal of the system.

The mechanism model diagram of single joint principle prototype is shown in Fig. 4. The model can be divided into six parts. Because the flexible cable moves at a low speed, high-frequency tremors are negligible, and the model of flexible cable is equivalent to a mass-spring-damper model. Motor movement cannot be completely passed to the slider. The displacement of slider 1 and the positive driving cable is determined by the detection value of grating ruler component 1, and the displacement of slider 2 and the negative driving cable is determined by the detection value of grating ruler component 2. The friction influence of the subsystems should be considered, and the gravitational effect of the gripper can be ignored since the gravity direction of the gripper direction and rotation direction are both vertical.

The dynamic equilibrium equations were established individually, as is shown below. When the system is in the original position, the flexible cables are in astatic state, and the joint angle and the displacement of sliders are zero. The relationship of the initial tensioning force of the positive driving cable and the negative driving cable are shown in Eq. (6), and F0 is the initial tension of the system. Fw10=Fw20=F0. The model of part 1 is shown in Eq. (7), where, im is the motor armature current, F1 is the load of the rope wheel, τl is the load torque, τm is the input torque, zi is the reduction ratio, rm is the effective radius of the rope wheel, Km is the torque constant, Jm is the effective moment of inertia of the DC motor, Bm is the viscous damping constant, φ¨ and φ˙ are the angular acceleration and angular velocity of the motor, respectively. τm-τl=Jmφ¨+Bmφ˙τm=Kmimτl=rmziFl. The model of the guide-slider and the idler component 1 (part 2) is shown below, where Fw1 is the cable tension that drives the gripper joint, m0 is the mass of slider 1, x¨1 is the slider acceleration,τf1 is the equivalent friction of part 2 and is composed of viscous friction and coulomb friction, fsv1 and fsc1 are the coulomb friction coefficient and the viscous friction coefficient, respectively, sgn⋅is the symbolic function, and x˙1 is  the slider velocity. F1-Fw1=m0x¨1+τf1τf1=τf1sv+τf1sc=fsv1x˙1+fsc1sgnx˙1. As the mass of the positive driving cable is very small, the viscous friction coefficient Bw=2ξw m1Kw, and the influence of viscous friction and mass can be ignored to simplify the model. The model of the positive driving cable (part 3) can be regarded as a spring model, as is shown in Eq. (9). Kw is the elastic coefficient of the positive driving cable and y1 is the displacement of one side of the positive driving cable, which is fixed to the gripper joint. Fw1=Kw1(x1-y1)+Fw10. The model of the spring (Sect. 4) is shown in Eq. (10). F2 is the spring tension, Ks is the elastic coefficient of the spring, and x2 is the displacement of the spring mobile side. F2=Ksx2+Fw20. Similarly to part 2, the model of the guide-slider and idler component 2 (part 5) is shown in Eq. (9), and the variables in Eq. (11) are defined as in part 2. Fw2-F2=m0x¨2+τf2τf2=τf2sv+τf2sc=fsv2x˙2+fsc2sgnx˙2. Similarly to part 4, the model of the negative driving cable (part 6) is shown in Eq. (12), and the variables in Eq. (12) are defined as in Eq. (9). Fw2=Kw2(y2-x2)+Fw20. The dynamic model of the gripper joint (part 7) is shown in Eq. (13), where, θ, τe, me, Je and r are the gripper joint angle, joint driven torque, joint effective mass, joint equivalent moment of inertia, and the effective radius of the cable tension, respectively. τef is the friction torque of the joint and is composed of the viscous friction torque and the coulomb friction torque τext is the external force moment, and le is the effective radius of the external force Fext. τe=Jeθ¨+τef+τextτe=Fw1-Fw2rτef=τefv+τefc=fefvθ˙+fefcsgn(θ˙)τext=Fextle. The relationship between the joint angle θ and y1, y2 is shown below, y1 is the displacement of the positive driving cable which is fixed on the gripper, and y2 is the displacement of the negative driving cable. y1=y2=rθ. Combining Eqs. (6), (11), (12), and (14), the two different expressions of Fw2 can be obtained, as shown below. Fw2=Kw2rθ-Kw2x2+F0Fw2=m1x¨2+fsv2x˙2+fsc2sgnx˙2+Ksx2+F0. Then the relationship between x2 and θ can be found as shown in Eq. (16). Equation (16) is the overall model of the joint angle estimator. The parameter identification and the practical application will be demonstrated in the experimental section. θ=m1Kw2rx¨2+fsv2Kw2rx˙2+fsc2Kw2rsgn(x˙2)+Ks+Kw2Kw2rx2.

The DOB regards the difference between the actual output and the output of the dynamic model (without external force) as a disturbance applied on the system. When an external force is applied, the equivalent disturbance includes not only the modelling uncertainties but also the external force information. Therefore, force estimation based on the DOB must find the external force information from the equivalent disturbance. The advantage of the DOB is that it does not need force sensors or other sensors to estimate the external force.

The DOB is designed to achieve the external force detection of the joint prototype system. The dynamics of a motor are described by the following equation. Jmφ¨=fd+Kmimreffd=τ′dis+Bmφ˙. Where: Jm, Bm, φ, imref and Km denote the inertia mass, damping coefficient, position response, current, and the force coefficient of the motor, respectively. fd denotes the motor load, and is composed of damping and disturbance, and the disturbance includes the external force. The basic block diagram of the DOB is shown in Fig. 5, where Jmn=Jm, and gdis is the parameter of the low-pass first order filter gdiss+gdis. If τdis1=0, the damping and disturbance are estimated using the low-pass first order filter as in the following equation. f^d=gdiss+gdisfd.

When the external force Fext is applied on the end of the gripper, the load of the rope wheel fixed on the motor is F1. If there is no external force applied on the end of the gripper, the load of the rope wheel fixed on the motor is F1′ and the relationship of F1 and F1′ is as shown below. F1=F1′+Fextr. The external force estimator for the single joint principle prototype system is designed as shown in Fig. 6. The output of the external force estimator is shown below. F^ext=gdiss+gdis(fd-f^d). Where fd′=F′^1rm/zi+B^mφ˙. To estimate the external force, the models of F′^1 and B^m must be known. The identification models of F′^1 and B^m are shown in the next part.

When the motor is running with no load,fd=Bmφ˙, and τdis1=0. The output of Fig. 5 is f^d=B^mφ˙, and Fig. 5 is the motor damping estimator. Parameter B^m is a constant that can be obtained by the least square method.

When the single joint principle prototype system runs without applied applied on the end of the gripper (Fext=0), let τdis1=B^mφ˙, and fd=F1′rm/zi+Bmφ˙. Then, the output of disturbance observer in Fig. 5 isf^d=F′^1rm/zi , and then F′^can be determined by the equation below. F′^1=f^dzi/rm. When the external force Fext is applied on the end of the gripper, we must get the estimated value of F1′ from the information we obtain on the state of the gripper contacting the environment. If the system is running without the application of external force on the gripper (Fext=0), the relationship of F1′ and F1 is F1=F′1 . The three models of F1′ were established by dynamic analysis of the single joint prototype system.

Model A: the model is established based on the dynamics analysis of part 2 and part 3. Combined Eqs. (6), (8) and (9), the expression of F1′ is shown below. F1′=m0x¨1+fsv1x˙1+fsc1sgnx˙1+Kw1x1-Kw1rθ+F0. Model B: the model is established based on the dynamic analysis of part 2, part 6, and part 7. Combining Eqs. (6), (8) and (12), (13) and (14), the expression of F1′ is shown below. F1′=Jerθ¨+fefvrθ˙+fefcrsgn(θ˙)+Kw2rθ+m0x¨1+fsv1x˙1+fsc1sgnx˙1+m1x¨2+fsv2x˙2+fsc2sgnx˙2-Kw2x2+F0. Model C: the model is established based on the dynamic analysis of parts 2, 4, 5 and 7. Combining Eqs. (6), (8), (10), (11), (13) and (14), the expression of F1′ is shown in Eq. (24). F1′=Jerθ¨+fefvrθ˙+fefcrsgn(θ˙)+m0x¨1+fsv1x˙1+fsc1sgnx˙1+m1x¨2+fsv2x˙2+fsc2sgn(x˙2)+Ksx2+F0. The actual value of F1′ and fd can be determined by the output of Fig. 5, as shown in the analysis above. The identification experiment and identified value of the model parameters are described in the next section.

In order to identify the parameters of the joint angle estimator, a micro magnetic encoder was used to detect the actual angle of only the gripper joint. The micro magnetic encoder has no effect on the control. The angle of the motor was used as the feedback signal. In other words, the motor was under closed-loop control, but the gripper joint angle was controlled by open-loop control, as shown in Fig. 8. The gripper angle θ can be obtained through the micro magnetic encoder, and x2 can be obtained using the grating ruler. Then, the parameters of the joint angle estimator model can be identified by the least squares method. The identification model is shown below. The meaning of the parameters ρs in Eq. (25) is the error vector.

Ys=Wsχs+ρsYs=θ,Ws=x¨2x˙2sgn(x˙2)x2χs=m1Kw2rfsv2Kw2rfsc2Kw2rKs+Kw2Kw2rT. Where, x2 is the position detected value of the Grating ruler component 2, x˙2 is the speed, and x¨2 is the acceleration, and sgn(x˙2) is the symbolic function.

Through the identification experiment, the parameters vector can be determined, as shown below. χ^s=0.08050.52250.0025264.3560T. The actual data, the identification result of the joint angle, and the errors are shown in Fig. 9. The average of the identification error is 0.2750∘ the maximum identification error is 0.8355∘, and the minimum of the identification error is -0.8321∘. These values indicate good accuracy of the joint angle estimator.

The performance of the general control system is shown in Fig. 10. There was significant error between the desired trajectory and the actual trajectory. The average error was 1.912∘, the maximum error was 4.134∘, and the minimum was 0.964∘. The performance does not the requirements for micro devices of precise operation.

In order to verify the performance of the joint angle estimator, a control system was designed and is shown in Fig. 11. The output of the joint angle estimator is the feedback signal of the position loop, the position control is a PD control and the speed control is a PI control. The parameters can be set through the servo drive. The parameters of the controller are shown below.

Positionloop:Kp1KD1=9814Speedloop:KvpKvi=10040. The desired trajectory and the actual trajectory are shown in Fig. 12. When the gripper joint is driving with no external force, the desired trajectory, the actual trajectory, and the estimated trajectory are almost the same. The average error between the desired trajectory and the actual trajectory was -0.150∘, the maximum error was 0.905∘, and the minimum error was -0.974∘. The average error between the actual trajectory and the estimated trajectory was 0.170∘, the maximum error was 0.715∘, and the minimum was -0.498∘. Overall, the control precision of the system is higher than the system without a joint angle estimator. In other words, the joint angle estimator works well and the necessary precision can be obtained.

To test the robustness of the control system, a complex trajectory was used to assess the performance of the control system and the joint angle estimator when an external force of 2.5 N was applied on the gripper. The experimental result is shown in Fig. 13. The average error between the desired trajectory and the actual trajectory is -0.105∘. The maximum error is 1.202∘, and the minimum is -1.271∘. The average error between the actual trajectory and the estimated trajectory is 0.076∘, the maximum error is 0.508∘, and the minimum error is -0.454∘. These values indicted excellent robustness of the control system and the joint angle estimator.

The motor was controlled without applied load on the motor output shaft, so that the actual output torque f^d can be obtained according to Fig. 5, and φ˙ can be obtained from the motor encoder. The relationship between f^d and φ˙ is f^d=Bmφ˙ can be determined and then the parameter Bm can be calculated by the least square method. The identification value of Bm is 1.6934×10-5 Nm rad-1. The identification experimental result is shown in Fig.14, and the verification experimental result is shown in Fig. 15. The result shows good precision of the identification of motor viscous damping.

The identification experiment was performed based on the joint angle closed-loop control by regarding the joint angle estimator output as the feedback signal. The control block diagram is shown in Fig. 11. The actual value of F′1 can be obtained as described in the analysis in Sect. 3.3. The values of the observation matrixes can be obtained by reading the grating sensors and the micro magnetic encoder. The three different models of F′1 are shown in Sect. 3.3, where the model C can be represented by the following expression, where x1 is the position detected value of the grating ruler component 1, x¨1,x˙1andsgn(x˙1) are acceleration, speed and symbolic function, respectively. The other models can similarly be written as expressions. The parameter matrixes of the three models can be obtained, and the identification results and the identification errors are shown in Fig. 16. Yc=Wcχc+ρc

Yc=F′^LWc=θ¨θ˙sgn(θ˙)x¨1x˙1sgnx˙1x¨2x˙2sgn(x˙2)x21χc=Je/rfefv/rfefc/rm0fsv1fsc1m1fvp2fcp2KsF0T.

The cable tension F′1 will have a larger error when the joint velocity is zero because the system has a hysteresis effect. The movement of the motor cannot be passed to the gripper immediately as the cable is not a rigid body. When the joint velocity is non-zero the identification results are trustworthy. The average errors of the three models are -0.0228, 0.0189, and -0.0063 N respectively. The identification precision of model C is higher than that of the others, so model C was adopted for further studies. The parameters and their standard variance are shown below. χ^c=χ^c1χ^c2χ^c3χ^c4χ^c5χ^c6χ^c7χ^c8χ^c9χ^c10χ^c11T=0.1975-9.8080.005-38.6467053.25938.99-44045.436669.019.53T%σχc^=32.29%27.3%2625%59.23%7.678%3.809%54.14%15.26%2.518%1.932%0.115%. As the velocity and acceleration of the system are small and the standard variance ofχc1, χc2, χc3, χc4, χc5, χc7 and χc8 are over 5 %, so the influence of these parameters can be ignored. The simplified model C is shown below. Yp=Wpχp+ρp,Yp=F′^LWp=sgn(x˙1)sgn(x˙2)x21χp=fsc1fcp2KsF0T. The parameters of the simplified model C can be obtained through the LS method. The identification result of the simplified model C is shown in Fig. 17, and the values of the parameters and their standard variance are shown below. Another experiment was used to verify the simplified model C, as shown in Fig. 18. The average error was 0.011 N. The above analysis illustrated the accuracy of the simplified model, suggesting the model can be used to estimate the external force applied on the gripper by the method presented in Fig. 6. χ^p=3.8795.027707.719.52T%σχ^c=3.025%2.658%1.787%0.115%.

The external force estimation strategy is shown in Fig. 6 in which the gripper was controlled by the method shown in Fig. 11. fd′=F′^1rm/zi+B^mφ˙ can be determined using the identification models in Sect. 4.1. External force applied on the gripper is shown in Fig. 19. The external force estimation experiments were performed by applying external forces on the gripper of different magnitudes and in different directions. The positive direction and negative direction external forces were applied at 0, 30 and -40∘, and external forces of 30, 50, 100, 200, and 250 g were tested. The experiment results are shown in Figs. 20 to 26.

The experimental results for a joint angle of 0∘ are shown in Figs. 20 and 21. Fig. 22a shows the experimental process when the external force is 2 N: (1) is the no-load region, (2) is the loading process, and (3) is the stable region. The no-load error range of the external forces estimation is about ±0.1 N, with high accuracy of the force estimator. The estimated accuracy can reach up to 94 % for an external force greater than 1 N, and the estimated accuracy can reach 82 % for an external force of 0.3 N. The external force has little influence on the joint angle control precision. The greatest error was 1.25∘ for an external force of 2.5 N.

An experimental process is shown in Fig. 22b for a joint angle of 30 and -40∘: (1) is the moving process, (2) is the adjustment process, (3) is the no-load region, (4) is the loading process, and (5) is the stable region. The dynamic no-load error is about ±0.15 N when external forces are applied on the positive direction. The dynamic no-load error is about ±0.2 N when external forces are applied on the negative direction. The static no-load error is about ±0.15 N, with good accuracy of the force estimator. The estimated accuracy can reach 95 % when the external force is more than 1 N, and the estimated accuracy can reach 84 % when the external force is 0.3 N. The external force has little influence on joint angle control precision. The greatest error was 1.4∘, when external force was applied to the gripper.