In the mid- to late-recovery stage, part of the motor-related nervous system of the patient has regenerated, and the affected limb has regained some motor function; however, it is still a great challenge for the patient to perform an independent training task. During this stage, the main therapy goals are twofold: (1) to maintain the existing range of motion and the degree of muscular activation and (2) to induce the patient to actively participate in or even lead the training task in order to further improve the patient's neurological function and gradually restore the motor ability of daily movements. Clinical experience has shown that maximizing the usage of recovered motor function is beneficial for patients to improve treatment efficiency and restore psychological confidence. Therefore, in this paper, a cooperative patient–robot control strategy for upper-limb rehabilitation is developed based on the principle of minimal intervention in order to facilitate human–robot interaction by stimulating patient initiative (Nef et al., 2007; Todorov and Jordan, 2013). It is required that the system not interfere with the rehabilitation training task if the patient can perform the expected training task actively; instead, maximum utilization of recovered motor function should be encouraged. However, assuming the patient is not capable of accomplishing the expected training task, the robot should provide the appropriate assistance to the affected limb in order to ensure the integrity of the training task.

A spatial force field is constructed in the 3D space around a predetermined horizontal training path, as shown in Fig. 1. During rehabilitation training, the actual path is likely to deviate from the desired path (e.g., the purple curve shown in Fig. 1). Thus, the position deviation d(t) from the actual position P(t)=[x(t),y(t),z(t)] to the reference position Pd(t)=[xd(t),yd(t),zd(t)] can be calculated. The spatial force field is divided into a patient-dominated area and a robot-dominated area. Note that RT and RN denote the boundaries of the patient-dominated area and the robot-dominated area, respectively. The actual motor capabilities of the patient can be estimated via the corresponding area. The position deviation d(t) can be represented as follows: 1d(t)=‖Pd(t)-P(t)‖2=[xd(t)-x(t)]2+[yd(t)-y(t)]2+[zd(t)-z(t)]2.

The actual end-effector position is within the patient-dominated area if the deviation satisfies the condition of d(t)∈[0,RT), as for point P(1) shown in Fig. 1, in which case the rehabilitation system judges that the patient's motor performance is good enough to actively complete the training task. Efficiency and accuracy are the most important two targets during this stage. The system gives little normal support in this area, and the patient can obtain adequate exercise. However, tangential force can be added to improve the patient's efficiency or as an appropriate push for the patient after a high-intensity training task.

If the position deviation continues to increase and satisfies the condition of d(t)∈[RT,RN), as for point P(2) shown in Fig. 1, the actual end-effector position is within the robot-dominated area; in this case, the rehabilitation system judges that the patient's motor performance is not satisfactory to complete the training task independently. The completeness of the training is the most important target at this stage. The system provides sufficient normal support, and this support increases exponentially as the position deviation increase until the centrifugal motion of the patient stops. This force can drive the patient's hand to return to the predetermined path.

It is necessary to mention that the controller parameters should be adjusted according to the different recovery stages and training requirements of patients. In the early stage of rehabilitation, the patient's affected limbs cannot complete the training task; therefore, some measures should be taken to help the patient complete passive training tasks – for instance, appropriately narrowing the patient-dominated area, increasing the normal support, applying greater tangential assistance and so on. On the other hand, patients who have recovered some motor function should be encouraged to actively participate in training; thus, it may be necessary to expand the patient-dominated area and reduce tangential assistance in order for the patient to gain more freedom of movement during the training task. In addition, if the patient's hand always moves back and forth between two areas, although it will not affect the stability of the system, the boundary of two areas still needs to be reasonably modulated to make it more suitable for the individual. Combining the above descriptions, the force-field control strategy is proposed as follows: 2Fx=FN+FT+FD,3Fθ=Kθ(θd-θ)-Dθθ˙,4Fassist=FxFθ. Here, FN∈R3×1, FT∈R3×1 and FD∈R3×1 represent the normal force, tangential force and damping item, respectively, and Fx∈R3×1 is the assistance force applied on the end-effector. Equation (3) describes an end-effector rotational impedance controller: a virtual rotational spring with rotational stiffness Kθ∈R3×3 is attached to keep the end-effector pose θ∈R3×1 close to the desired pose θd∈R3×1. The rotational velocity θ˙∈R3×1 is damped with dissipating element Dθ∈R3×3, and Fθ∈R3×1 is the assistance wrench applied on the end-effector. Control models for each item in the force field are as follows: 5FN=KN1+e-Ns(d-RN)n,6FT=KTRT-dRTt,d<RT0,d≥RT,7FD=-KDx˙,8n=Pd-Pd.

Here, n∈R3×1 (in Eq. 5) represents the direction of normal force, which can be calculated by Eq. (8); KN∈R3×3 is the coefficient of normal force; KN=αKNt-1+βD, where α is a forgetting factor, and β is a gain factor; KNt-1 represents the normal force coefficient at the last sample time; D=diag{[dx,dy,dz]} indicates the real-time position deviation of the end-effector in three directions; KT∈R3×3 (in Eq. 6) denotes the tangential coefficient; t∈R3×1 is the direction of the tangential force, which is always in the same direction as the motion; KD∈R3×3 (in Eq. 7) represents the damping coefficient; and x˙∈R3×1 is the translational speed of the end-effector. The variation characteristics of the tangential and normal force around the predetermined path are shown in Fig. 2.

The analytical Jacobian matrix J(q)∈R6×7 of the robot is donated by Eq. (9); here, J(q)x∈R3×7 maps the joint velocities q˙∈R7×1 to the translational end-effector velocities, and J(q)θ∈R3×7 maps q˙ to the rotational end-effector velocities: 9J(q)=J(q)xJ(q)θ.

For the force-field controller and end-effector rotational impedance controller, the desired control torque τx∈R7×1 and τθ∈R7×1 can be computed by Eqs. (10) and (11), respectively. On this basis, robot motion in null-space is constrained (Hermus et al., 2022), and the desired null-space control torque τnull-space∈R7×1 is donated by Eq. (12). 10τx=JT(q)xFx11τθ=JT(q)θFθ12τnull-space=Nnull(Kq(qd-q)-Bqq˙) Here, Nnull∈R7×7 is the null-space projector and is defined as follows: Nnull=I-JT(q)(JT(q))#. Kq∈R7×7 is joint-space stiffness, Bq∈R7×7 is joint-space damping and q∈R7×1 is a real-time joint position with the nominal joint position qd∈R7×1 being constant throughout the trial. The desired assist torque can be represented as follows: 13τassist=τx+τθ+τnull-space. Figure 3 shows the overall block diagram of the proposed force-field-based robot rehabilitation system. The left side depicts the force-field control strategy mentioned in this paper, and the right side shows patient–robot interaction.

EULRR is designed to mimic the relative position of the human arms for more effective rehabilitation training. Figure 4 describes the structure of EULRR, which includes a pair of industrial manipulators, an external monitor and the body with an electrical module, a pair of manipulator control cabinets, and a control computer.

During the test, the subject sat inside the EULRR and kept their torso still, as shown in Fig. 5. The subject's hand was connected to the end of the robot via a handle, with the assumption that the subject's left upper arm was the affected side. To verify the effectiveness of the force field in different areas, the subject had to cross two areas in one single experiment; therefore, a state of muscular spasticity (MS) was artificially constructed, which required the subject to complete the training with a large deviation. Subjects were required to intentionally manipulate the robot end-effector, circulating along the predetermined path for seven laps in every single experiment. During laps 2 to 4, the subjects were asked to complete the training task as well as possible in a muscle normal (MN) state; however, during laps 6 and 7, the training task was performed in the MS state. It is important to note that, in this paper, position deviation is the criterion used to distinguish acceptable or unacceptable training performance.

The experiments were designed under the principle of control variates. The fixed parameters of all of the sub-experiments are shown in Table 1. The first experimental group (E1 in Table 2) contains 12 conditions, named T1 to T12, and aims to investigate the effect of different combinations of KT and KD on the force-field control performance. The second experimental group (E2 in Table 2) contains three conditions, named T13 to T15, and aims to investigate the effect of different combinations of α and β on the force-field control performance. The third experimental group (E3 in Table 2) is the control group and has only one condition, named T16. The control group T16 is performed before all of the other groups.

The system provides the subjects with visual guidance, developed based on Unity-3D, as shown in Fig. 6. The green dot on the interface is the real-time position of the end-effector, the arrows point out the real-time directions of the tangential and normal forces, the yellow circle is the top view of the predetermined path, the yellow line is the front view of the predetermined path, and the completed lap of the subject is recorded in the upper left-hand corner. The subject moves clockwise.

The experimental data were all processed with MATLAB and SPSS software. The real-time position of the end-effector was calculated according to the robot forward kinematics. The position deviation of the end-effector was calculated using Eq. (1), and the assistance force of the robot was calculated using Eq. (14). 14Fassist=Fx_x2+Fx_y2+Fx_z22

The actual path was visualized along with the predetermined path in order to intuitively point out the difference between robot-assisted movement and free movement, and the average assistance force was displayed along with the end-effector position deviation in order to intuitively discover their relationship. The abrupt engagement of the robot controller at the start of each experiment induced transient behavior in the robot end-effector motion, as the task was not critically damped. To eliminate possible transient behavior from the data analysis, the first revolution in each of these trials was discarded, and the fifth revolution was discarded as well because it recorded the transition behavior between two states, which was meaningless to the experiment.

The actual path of the hand of the first subject (S1) under condition T14 and condition T16 are shown in Figs. 7 and 8, respectively. The red curves are conducted in the MN state, and the blue curves are conducted in the MS state. The actual paths conducted in the two states are significantly different: under condition T14, the average respective deviations in the MN and MS states are 6.51 and 19.35 mm, whereas those under condition T16 are 23.05 and 34.28 mm. By analyzing all five subjects' average deviations between the two states using a paired t test, significant differences between two states were uncovered, as shown in Table 3, which proves the feasibility of the MS state.

Even though the 3D view in Figs. 7 and 8 clearly shows the difference in the actual path between robot-assisted motion and the free motion, there is no significant distinction between them in the xoy view because the graphical guidance works, but it is difficult for the subject to balance the motion in both the xoy plane and the yoz plane at the same time, which results in greater deviation in the z-axis direction when the subjects are experiencing free movement. This is evident from the xoz or yoz view.

The experimental results containing the average assistance force and deviation of the tests conducted by S1 are shown in Fig. 9. The effect of different combinations of control parameters can be revealed. The red line depicts the variation in the average deviation, while the blue line indicates the variation in the average assistance force. It can be noted that the presence of a force field causes the end-effector deviation to be smaller than that under free motion, regardless of whether the subject exercises in the MN state or the MS state. For each state, the results are statistically analyzed using a paired t test in order to compare the results between conditions. The threshold for the p value is selected to be 5 % for all tests. The results of the paired t test are shown in Table 4. If the t test result is shown with a “–”, it means that the mean deviation of the two sets of data is similar; otherwise, the percentage decrease is shown in the table with the p values given in parentheses. The main findings with respect to the paired t test results are as follows:

For the two states, conditions T1–T15 and condition T16 present significant differences in their mean deviation: all conditions show great improvements compared with condition T16, which proves that the force field can help subjects improve their training performance. Thus, the feasibility of force field for rehabilitation training is verified.

In experimental group E1, the variation in the assistance force and the deviation show the same trend when the characteristics of the normal force are controlled. However, there are still some cases where the assistance forces have opposite trends to the position deviations (e.g., condition T3 in Fig. 9a), and this can be explained by the fact that the average deviations in the MN state are small, with the contribution of the tangential force to the assistance force being much larger than that of the normal force. Nevertheless, when the subjects perform the tasks in the MS state, the position deviations are larger overall, and the effect of tangential force is smaller, resulting in the same trend between the average deviations and assistance forces. The above analysis indicates that the force field can respond effectively to the performance of the subject by applying time-varying and appropriate assistance force.

In experimental group E1-1, the average deviations have a different trend from KT when KD remains constant. This conclusion is also applicable to the comparison between different experimental groups – for instance, condition T4 in E1-2, condition T7 in E1-3 and condition T10 in E1-4 also show the same result. In contrast, for groups E1-2, E1-3 and E1-4, the value of KT remains constant, but the average deviations do not grow with an increase in KD. These two points combined with result (2) from the paired t test (shown above) indicate that KT and KD do not play a decisive role in the position constraint. However, as an increase in KT makes the subject's hand move faster, it is necessary to set an appropriate KD as a limitation for movement.

In experimental group E2, the values of KT and KD remain constant, and the variation characteristic of normal force is changed by adjusting the value of β. As the value of β increases, the average deviations gradually decrease, and the average assistance forces increase as well or are almost the same. In the other words, the subject is provided with a larger assistance force at the same position. Therefore, the constraint capacity of the force field is enhanced. The above conclusions combined with result (3) from the paired t test (shown above) provide evidence that the controller parameter should be adjusted in real-time according to the patient's motor performance.