The advantages of elastic joints are first outlined by Pratt and Williamson’s research (Pratt and Williamson, 1995), where a proportional-integral-derivative (PID) control scheme was proposed. Similarly, proportional-derivative (PD) controllers with on-line gravity compensation are considered in De Luca et al. (2005), Zollo et al. (2007). The global asymptotic stability of these control laws is demonstrated via Lyapunov’s argument and La Salle’s theorem. Another alternative approach is based on robust controllers with a disturbance observer (DOB) (Kim and Chung, 2015). In Talole et al. (2010), a combination of a feedback linearisation-based controller for the trajectory tracking problem and an extended state observer for uncertainty and states estimation is considered. Feedback linearisation and robust integral of sign of error (RISE) methods for controlling position are proposed in Yin et al. (2016). Specifically, the dynamics of the actuator is first feedback linearised, then a RISE method is applied to adapt the system model to uncertainties. A model reference adaptive control approach is implemented in Losey et al. (2016) to adapt to uncertainties in system parameters, while the adoption law is demonstrated using Lyapunov’s theory. In Pérez-Ibarra et al. (2017), a H ∞ force controller is used to deliver precisely the force required by a PD position controller on the outer loop. In Kaya and Çetin (2017), an adaptive state feedback regulator and a conventional PI controller is adopted to control a rotary series elastic actuator. The adaptive state feedback regulator define the transient behaviour of the SEA, then, the tracking error is reduced by the PI controller. In Penzlin et al. (2019), a clutched parallel elastic actuator (CPEA) with an iterative learning control loop and a cascade control loop is proposed. The cascade control loop is used to control the position output of a BLDC motor. The iterative learning control loop is used to control the output trajectory of the actuator. To the best of our knowledge, an adaptive control aiming at both stabilising the system as well as dealing with uncertainties to eliminate the influence of external force/torque has not been released yet.

Elastic elements play a very important role in elastic joints. They are responsible for absorbing unexpected collision and storing elastic energy. The elastic elements can be springs and can have different shapes (as shown in Figure 2). Springs in elastic joints can be classified into torsion spring and extension/compression spring. In (Pratt et al., 1997, 2002; for; Human and Cognition, 2020), a pair of extension/compression springs are used, as shown in Figure 2A. In (Sanfilippo et al., 2019a), two springs are bent to fit into a compact housing, as shown in Figure 2B. Beside conventional springs, there are some custom designs of the elastic element. Some examples are shown in Figure 2.

To amplify the torque and reduce the speed of a motor, gearing systems are adopted. Besides the common spur gears, some special types of gear are also utilised in elastic joints. Strain wave gearing, also known as harmonic gearing, can be applied, e.g., (Negrello et al., 2015; Wang et al., 2017; Sergi et al., 2012). This mechanism consists of three basic components: a wave generator, a flex spline, and a circular spline. The advantages of the harmonic gearing mechanism are no backlash, high compactness, lightweight, high gear ratios in a small volume, and coaxial input and output shafts. In the research of Meng (Wang et al.,2015; Wang et al., 2017), a nonlinear series elastic joint with variable stiffness is presented. There are two motors and a harmonic gearing mechanism in this design. One motor is connected to the wave generator, while the other motor is used to change the stiffness preset, and the link is attached to the flex spline. Planetary gearing system is also utilised in elastic joints, e.g., (Lee and Oh, 2016; Plooij et al., 2016). The advantages of the planetary gearing mechanism are compactness, high efficiency, low backlash, high torque density (i.e., high torque-to-mass ratio), and coaxial input and output shafts. In Lee and Oh (2016), good control performance is achieved while the size is still compact. The motor shaft is connected to the Sun gear, the torsional spring is connected to the ring gear, the load is attached to the carrier and the planet gears combine these three parts. One side of the spring and the motor is attached to a fixed base. In Plooij et al. (2016), a BIdirectional Clutched Parallel Elastic Actuator (BIC-PEA) is introduced. The mechanism contains a planetary differential, a torsion spring, and two brakes. It is stated that BIC-PEA design can save up to 65% of the energy consumption in some specific tasks.

The schematic diagrams of the elastic actuator are illustrated in Figure 3. There are two gears with gear ratio N = N l / N m , where N l and N m are the number of teeth for the load and the motor gear, respectively. The torques shown in these diagrams are motor torque ( τ m ), spring reaction torque ( τ s ) and external torque ( τ e x t ). Figure 3A illustrates the system when there is no external force/torque, while Figure 3B illustrates the system affected by an external action. The compression of the springs on the real prototype is also shown on Figure 3C.

The whole system is affected by disturbances from the motor ( d m ) as well as from the load ( d l ). These parameters represent external noise and noise caused by electromechanical systems (e.g., vibrations of the mechanical parts, voltage spike in electronic components, etc). By denoting the motor angular position as θ , the load angular position as q, the rotor inertia as J m , the motor damping coefficient as D m , the stiffness coefficient of the spring as K s , the spring damping coefficient as D s , the load inertia as J l , and the load damping coefficient as D l , the mathematical model of the elastic joint system is obtained: d m + τ m − N − 1 τ s = J m θ ¨ + D m θ ˙ , (1) τ s = K s ( N − 1 θ − q ) + D s ( N − 1 θ ˙ − q ˙ ) , (2) d l + τ s + τ e x t = J l q ¨ + D l q ˙ . (3)

Eq. 1 shows the relationship on the motor-side between the motor torque, the spring torque and the motor angular position. The spring torque, τ s , is obtained by Eq. 2. The interaction between the spring torque, the external torque and the load angular position is illustrated by Eq. 3.

The proposed control algorithm diagram is presented in Figure 4. The objective of this work is to develop a controller that can track the desired load angular position when considering external disturbances on the load and uncertainties in system parameters. To achieve this objective, two separate types of controllers are utilised for the motor-side and the load-side, respectively. For the load-side, we propose using a Fuzzy PI Controller (FPIC) to reduce the effect of external disturbances on the load. The considered external disturbances are the forces/torques caused by undesired collisions when operating in unknown environments. The output of the FPIC is used as the desired angular position of the motor. For the motor-side, a Model Reference Adaptive Controller (MRAC) is used to cope with uncertainties in system parameters. The uncertain system parameters could be the inertia of the load or the stiffness of the spring. The idea is inspired to a previous research of (Losey et al., 2016). The advantage of combining the FPIC and the MRAC controllers is the possibility of achieving independence with respect to imprecise system parameters. In previous works (Losey et al., 2016), the motor angular position is prioritised more than the load position because the target is force control. The contribution of the paper is that a robust position control algorithm for the load position is proposed. In addition, a better spring model is given with the damping coefficient added.

In this section, the mechanical design of the elastic joint and the robot arm are described. The mechanical components of the elastic joint are shown in Figure 6A. A 3D view of the same parts is illustrated in Figure 6B, while the assembled view of the actuator is illustrated in Figure 6C–E. The mechanism can be separated into load side and motor side. The motor side mechanism includes an 8 mm metal shaft, an 8 mm metal flange, a flange extension, a cover, and a driving joint (component 9, 8, 7, 6, 5 in Figure 6A, respectively). The load side mechanism includes an 8 mm metal shaft, a cap, an 8 mm metal flange, and a compliant joint (component 1, 2, 3, 4 in Figure 6A, respectively). The shafts are connected to the other parts by the flanges. The flange on the motor side is connected to the flange extension and the flange extension is connected to the cover and the motor side driving joint. The motor side driving joint has space to place two springs with 8 mm outer diameter. When the motor rotates, the springs are compressed. This will create an elastic force on the compliant joint. The load side shaft is connected to the joint through the flange. The shaft is kept in place by a cap. In our work, two extension/compression springs are chosen because they form a symmetrical configuration. In addition, the extension/compression type of spring is used instead of torsion spring because it is more durable and can bear a larger force.

Based on the newly introduced actuator, the design of the 2-DOF elastic robot arm is shown in Figure 7. The joints of the robot arm are revolute joints. The motor and the elastic joints are connected by pulleys and belts. The transmission ratio from the motor to the actuator is 4:1 (the pulley of the motor has 20 teeth and the pulley of the actuator has 80 teeth).

The simplified model of the robot arm is shown in Figure 8. The Denavit-Hartenberg table is shown in Table 2. The angles are in radian and the lengths are in meter. From this table, the corresponding transformation matrices are formed, as shown in Eq. 30. T 1 2 = [ c o s ( q 1 ) 0 s i n ( q 1 ) 0 s i n ( q 1 ) 0 − c o s ( q 1 ) 0 0 1 0 l 1 0 0 0 1 ] ,       T 2 3 = [ c o s ( q 2 ) 0 − s i n ( q 2 ) l 2 c o s ( q 2 ) s i n ( q 2 ) 0 c o s ( q 2 ) l 2 s i n ( q 2 ) 0 − 1 0 0 0 0 0 1 ] . (30)

In this section, the hardware used in the robot arm is presented. The motor used in the robot arm is the GA25 DC motor (XYTmotor, 2020). It is a 12 V DC motor with a 4 mm shaft and a two channels encoder attached. The gearbox transmission ratio is chosen as 217:1. The encoder for the load side is the AMT-20 encoder (devices, 2020) from CUI Devices. It has incremental and absolute encoders in a compact package. The incremental encoder has three channels (A, B, and Z). The absolute encoder has 12 bits of absolute position information with Serial Peripheral Interface (SPI) communication. The microcontroller used is the STM32F446RE on the STM32 Nucleo board (Stmicroelectronics, 2020).

The software design is described in this section. The concept of modularity is applied to the control architecture. Each joint is independent, being controlled by a self-reliant slave controller, which directly communicates with the master controller running on a personal computer (PC). Each slave controller communicates with the PC through a Universal Serial Bus (USB) port on the board. Figure 9 shows the diagram of the control architecture. Each STM32 board has timer peripherals, which are used to capture the encoder signals. The timers are also configured to be Pulse width modulation (PWM) output to control the motors through H-bridges (nhatbon, 2020). The control/update rate of the controller is 100 Hz (10 milliseconds). The presented framework is a multi-layer architecture that includes the following layers: High-level control layer: it considers the control of the overall manipulator, it includes kinematics according to the corresponding DOFs. Low-level control layer: it considers the control of the single joints. This makes it possible to use the proposed approach for robotic arms with different DOFs.

The response of the load-side system in normal condition is illustrated in this section. Step and sine responses are shown in Figure 10. In these figures, blue dashed lines are desired inputs and red solid lines are load angular positions. The unstable stage at the beginning of the simulation is the learning phase of the MRAC. The results show that the proposed controller has relatively fast response and small error.

The response of the motor-side system in normal condition is presented in this section. Step and sine responses are shown in Figure 11A,B , respectively. The reference model and the MRAC parameters are shown in Table 4tbl4. The system parameters are modified based on the adaption law (Eq. 26) to reduce the error between reference model and real motor response. The adaption law is designed using Lyapunov’s approach. This has an advantage: arbitrary large values of the learning rate coefficient γ can be used (Åström and Wittenmark, 2013).

The damping coefficient ξ of the reference model is chosen to be critically damped ( ξ = 1 ) because there are no oscillations or overshoots in this configuration and the system returns to equilibrium in minimum time. However, Figure 11A shows that there is a small overshoot in the reference model graph. This is caused by the integrator in the FPIC, not by the reference model in the MRAC, and could be reduced by appropriately adjusting the K p coefficient. The rising time of the reference model is calculated by using the approximation formula: 5.83392 ω n .

In this section, the effect of the disturbance on the elastic joint and the efficiency of the proposed algorithm is demonstrated. The disturbance signal affecting the load side is a sine wave signal with an amplitude of five radian and a cycle time of 0.5 s. This signal is applied to each joint. This is equivalent to the vibrations often occurring in elastic joints. The sine response of the load-side system and the motor-side system is shown in Figure 12A,B, respectively. The motor control voltage is shown in Figure 12C. In the proposed two-loop controller, the FPIC is adopted on the load side to reduce the influence of external disturbances and the MRAC is adopted on the motor side to deal with uncertainties. When there is an external torque, the desired motor angular position is adjusted by the FPIC, so that the error between the desired load angular position and the real load angular position goes to zero. As shown in Figure 12B, the motor angular position changes when the external torque appears and the influence of this external torque is eliminated.

To show more evidently the effectiveness of the proposed control algorithm, a standard PID controller is applied and its results are compared. As shown in Figure 13A,B, the responses of the PID controller and the FPIC-MRAC controller are similar in normal condition. However, as shown in Figure 13B and Figure 12A, the response of the FPIC-MRAC controller is significantly better than the PID controller in the presence of external disturbance.

The impact of human-machine interaction on the elastic joints is presented in this section. The load side response and the control voltage are shown in Figure 14. The simulated human-machine interaction of joint 1 appears from the second 20 to 30 and from 40 to 45. The simulated human-machine interaction of joint 2 appears from the second 10 to 20 and from 30 to 40. The collected data outline that despite being affected by the disturbance, the controller can adjust the voltage to keep the system stable.

In this subsection, experiment results related to the control of the real 2-DOF robot arm with elastic joints in the normal condition are presented. The normal condition is when the robot arm is not disturbed by the external torque. The joint angle response and control voltage are demonstrated. There are two input signal types: sinusoidal and square waveforms. The sinusoidal waveform input has an amplitude of 0.5 rad and a cycle time of 8 s. The square waveform input has an amplitude of 0.5 rad and a cycle time of 8 s. The data is collected within 50 s.

Figure 15; Supplement Figure S1 show the angle response and the control voltage of joint 1 and joint 2, respectively, in normal condition with the sinusoidal waveform input. The figures demonstrate that the decentralised control algorithm works as expected. As shown in Figure 15A, the absolute value of the positive peak of the motor angle is larger than the negative peak. This is due to a small manufacturing defect in the actuator: the spring on one side is stiffer than the spring on the other side. Despite this fact, it is clear that the controller still tracks the desired load angle.

The angle response and the control voltage of joint 1 and joint 2 are shown in Supplement Figures S2, S3, respectively, in normal condition with the square waveform input. These figures demonstrate that the decentralised control algorithm also works well in this situation. The error decreases to smaller than 10° after just a few seconds.

In this subsection, the experiment results of the real 2-DOF robot arm with elastic joints in presence of disturbances are presented. The joint angle response and control voltage are demonstrated. The sinusoidal waveform input has an amplitude of 0.5 rad and a cycle time of 8 s. The data is collected within 50 s. The external disturbance is created by randomly holding the robot arm by hand while it is operating. This imitates the unexpected collision with a human operator while coexisting in a shared work-space.The angle response and the control voltage of joint 1 and joint 2 in disturbance condition are shown in Supplement Figures S4, S5, respectively. These figures demonstrate that the decentralised control algorithm can stabilise the system when there is unexpected torque. In addition, if the absolute value of the deviation between the motor angle and the load angle is larger than 30° (0.523,599 radians), the motor will stop. This value can be tuned appropriately, which means that safety can be guaranteed.

In this subsection, trajectory tracking experiment results of the elastic robot arm in the normal condition are demonstrated. A sequence of images of the experiment is illustrated in Supplement Figure S6. The initial position is when q 1 = 0 , q 2 = 3 π 4 and the end position is when q 1 = π , q 2 = 7 π 6 . The desired trajectory and the response of the robot arm are shown in Supplement Figure S7A. It is clear that the end effector of the robot can be following the desired trajectory. In details, Supplement Figure S7B,C, show the responses of joints 1 and 2, respectively. The tracking error is relatively small.

In this subsection, trajectory tracking experimental results of the elastic robot arm when there is an unexpected torque are demonstrated. A sequence of images from the experiment is shown in Supplement Figure S8. The initial position is when q 1 = 0 , q 2 = 3 π 4 and the end position is when q 1 = π , q 2 = 7 π 6 . As shown in the third image of the sequence (Supplement Figure S8), the unexpected torque is produced by the collision with a human hand. The desired trajectory and the response of the robot arm are shown in Supplement Figure S9A. Obviously, there is a section where the end effector of the robot arm deviates from the desired trajectory. It corresponds to the period when the arm is affected by undesired torque. Supplement Figure S9B,C illustrate the responses of joints 1 and 2, respectively. As shown in these figures, when the collision occurred from second 7 to second 12, the robot arm tries to hold its position to mitigate the damage. After that, when the robot arm is released, it gradually converges to the end position.