3.1. Platform Design
In this paper, we adopt a self-designed 7-DOF humanoid manipulator concerning the arm of a male adult (1.7 m in height), shown in
Figure 3. The robotic arm consists of seven motors, including seven rotational degrees of freedom. The shoulder joints (No.1–3) and elbow joint (No.4) use the planetary reduction joint actuator QDD-PR60-36 with a reduction ratio of 36. The wrist joints (No.5–7) use the planetary gear actuator QDD-NE30-36. The performance parameters of the motor are shown in
Table 1. The torque and rotational speed are close to the daily activity data of the human arm [
20,
21]. The brachium and forearm are made of carbon fiber material with lengths of 274 mm and 265 mm. A full arm weight of 5 kg is close to the average human data. The Inspire-RHF-6-BXF dexterity hand is used to realize the grip** action of the steering wheel. It contains six DOF in whole, two degrees of freedom for the thumb, and one degree of freedom for each of the other four fingers.
To enlarge the operating space of the robot in front, the first joint is installed oblique 45 degrees, which makes the robot arm inverse solution more difficult but it can be solved by changing the definition of the base coordinate system. In addition, for the need to carry large loads, crossed roller bearings are used at the non-flanged output of the motor to improve the stiffness of the robotic arm. Moreover, joints 5, 6 and 7 are orthogonally arranged to imitate the human wrist and achieve some actions such as the ’folding wrist’ and ’buckling wrist’. Joint 5 is set forward to the elbow to improve wrist mobility, which is different from most 7-DOF robotic arms. It makes the end motion close to human and reduces the possibility of wrist singularity in the forward working space. By combining the movements of these seven joints, this manipulator can replicate the intricate and precise movements of a human arm.
3.2. Kinematic and Dynamic Modeling
As for kinematic solutions, modeling the arm’s structural body is required. Usually, the direct kinematic modeling process of the humanoid manipulator is to derive the end-effector in Cartesian space by calculating the rotation angle of each joint, while the inverse kinematics is to obtain the rotation angle of each joint by deducing the end-effector in Cartesian space in the reverse direction [
22]. The arm configuration determines the relationship between the humanoid manipulator joint space and the end motion in Cartesian space and is represented by the D-H parameter [
23]. The zero position state of the redundant manipulator and the corresponding D-H parameters are shown in
Figure 4a and
Table 2.
The inverse kinematic solution of the humanoid manipulator plays an important role in robot control. The Jacobi matrix represents the relationship between the joint rotation velocity and the end tool actuator motion velocity, while the Jacobian of a redundant manipulator is a non-square matrix, and its generalized inverse matrix is generally obtained by Singular Value Decomposition (SVD). The speed of N joints’ rotation is denoted as
, the 6 DOF motion of the end-effector is indicated as
, and they are related by the Jacobi matrix, as expressed in the equation below
The Jacobi matrix of the redundant manipulator is rectangular, represented by its pseudo-inverse as
[
24]. Depending on the humanoid manipulator’s specific configuration, its Jacobian can be treated as a general linear equation system solution problem, and under the condition that the Jacobi matrix rows are of full rank, the solution is as follows
Dynamics modeling is required because of the interaction between the humanoid manipulator and the steering wheel. The kinetic equations of the humanoid manipulator are derived from the Newton-Euler equation as follows, where
is the mass matrix of the humanoid manipulator,
is the centrifugal force and Gauche force vector,
is the gravity vector, and
is the external output torque of the humanoid manipulator [
25]
The relationship between the force
F acting on the actuators at the end of the humanoid manipulator and the torque
applied to the joint can be related by the Jacobi matrix, such as
, and the final dynamics modeling is expressed by the following equation
3.3. Admittance Controller Design Based on Parameters Fuzzification
This section focuses on the task of steering wheel operation by proposing an admittance controller based on parameters fuzzification, which consists of a trajectory tracking strategy to maintain circular motion through velocity control in Cartesian space [
4]. An admittance controller supple strategy based on force sensors at the end-effector, and parameters fuzzification to improve the performance to adapt to different scenarios is used.
For the first question in the previous section, the end-effector is required to rotate while being able to accurately track a circular motion trajectory to always keep one axis pointing to the center of the circular motion, and this motion constraint
is described under Cartesian space as follows
where
is the angular velocity of the rotational motion of the end-effector in the x-axis direction under the end coordinate system E,
and
are the velocity of the translational motion of the end-effector in the x-axis and y-axis under the base coordinate system B, respectively, R is the radius of circular motion, and
determines the rotational speed of the manipulating steering wheel. Since only the operation performed by the humanoid manipulator after grip** the edge of the steering wheel needs to be considered, this decision can be adapted to different types of humanoid manipulator end-effectors.
For the second question, it is necessary to design the admittance controller to realize the active compliance of the humanoid manipulator. Impedance/admittance control was first proposed by HOGAN [
26] in 1985, and the goal of this method is to achieve the ideal dynamic relationship between the robot end position and end contact force, which is highly adaptable and can be applied to the design of robot motion control in both free space and constrained space. The following equation represents its second-order model
where
is the inertia coefficient matrix,
is the dam** coefficient matrix, and
is the stiffness coefficient matrix.
is the 6-dimensional force generated by the admittance model to the environment,
x and
are the actual and desired poses, respectively [
27]. As mentioned above, the end-effector needs to track a given circular trajectory, so the positional deviation
generated by
alone is expressed as follows
The velocity deviation
is obtained by differentiating
, and
is differentiated to obtain the acceleration deviation
. The admittance model is further rewritten as the following equation
Combining the external force
which acts on the actuator at the end of the humanoid manipulator and the output
of the admittance model, the final velocity control quantity
in Cartesian space is given by the following equation
The flow chart of the humanoid manipulator compliant control is shown in
Figure 5.
To improve adaptability at different operating distances [
28], the
,
and
parameters of the admittance controller are fuzzified. Fuzzy control is a rule-based control that directly adopts language-based control rules and establishes the map** relationship between controlled quantities and semantic rules through the control experience of field operators or the knowledge of relevant experts [
29].
Figure 6 shows several pictures of different distance limits of the humanoid manipulator body from the steering wheel, corresponding to the degree of joint4.
The construction of a Fuzzy Inference System (FIS) is based on the Takagi-Sugeno model [
30]. The change of
away from 90° and its rate of change are used as fuzzy control inputs, which are represented by
respectively. Because of the task space being in front of the manipulator, only
is considered.
X contains membership functions as zero (Z), positive small (PS), positive medium (PM) and positive big (PB).
Y contains membership functions as negative big (NB), negative small (NS), zero (Z), positive small (PS), and positive big (PB). The fuzzy control output includes parameters of the admittance controller, which contain membership functions as low (L), medium (M), and high (H).
use the relevant fuzzified rules, which are shown in
Table 3.
The farther away the manipulator is from the steering wheel, the closer it reaches the limit attitude of the manipulator, and the more likely it becomes a singularity. In this case, the system will be unstable if the admittance controller produces the output due to vibration or steering wheel slewing torque that pulls the end-effector to the limit state [
10]. It can also be seen from
Figure 6 that the angle of the humanoid manipulator elbow joint (motor 4) has a linear relationship with the operating distance, so the fuzzy rule can be set by this experience to adjust the corresponding inertia, dam**, and stiffness parameters, with the initial posture elbow joint angle of 90° as the dividing line.
Figure 7 shows the flow chart of our control method.
where
,
and
conform to the following rule,
represents the angle of motor 4 at the initial attitude of the motion.
is determined by the deviation of
from the minimum value of the current interval at the initial attitude, such as