Interior wall finishing design model incorporating motion redundancy optimization and active compliance control algorithm

Abstract

Faced with the rising labor costs and low construction efficiency in the construction and decoration industry, there is an urgent need to introduce automation technology to reform construction methods. Therefore, a flexible macro and micro decoration robot is designed, integrating motion redundancy optimization and active compliance control strategies. The gradient projection method is used to optimize the accuracy of the robot arm’s motion trajectory, while the active supple control is used to improve the wall surface adaptability to enhance the stability and accuracy of the finishing actuator system’s force-position tracking. The results showed that the macro/micro robotic arm was accurate in tracking the position and attitude of the trajectory, with only 0.1° in the β-Euler angle. The joint motion of the robotic arm and the driving speed of the micro robotic arm were maintained within reasonable limits. Under the premise of ensuring accurate force and position tracking as well as system stability, the stiffness coefficient should not exceed 10 to avoid frequent small oscillations during tracking and instability due to excessive stiffness. The redundant motion control algorithm and active compliance controller based on impedance control studied have improved the adaptability and accuracy of the robot arm. The designed method can achieve precise wall finishing control, which makes a significant contribution to improving construction efficiency and quality, as well as ensuring worker safety.

Keywords

interior decoration supple control redundancy motion robotics

Introduction

In modern society, robots are increasingly taking over traditional physical labor, particularly in the construction industry. These robots not only alleviate significant labor demands but also markedly enhance work efficiency.¹ In the construction sector, wall finishing plays a crucial role, as it not only improves the functional and aesthetic value of buildings, but also contributes to environmental beautification. Traditionally, wall finishing has relied heavily on manual labor, which dominates both the time and cost aspects of construction projects. In contrast, wall finishing robots can result in an estimated 85% reduction in labor costs while boosting productivity by a factor of 10 to 15 times, thereby minimizing time and overall expenditure.^2,3 Therefore, using robots for high-quality wall finishing has become increasingly important. Nowadays, wall plasterers not only perform basic finishing tasks, but also supervise automatic coating thickness control and surface treatment.⁴ However, despite these technological advancements, wall finishing robots may lack the flexibility of manual labor in adapting to non-standard wall surfaces or specialized finish designs. In some older buildings or unique projects, the integration of new technologies with existing processes can encounter challenges related to technology adaptation and system compatibility.⁵ Therefore, faced with challenges such as rising labor costs, low construction efficiency, and difficulty in meeting increasingly complex design requirements in the building decoration industry, a new type of decoration robot is designed, which combines macro robotic arm and micro robotic arm and a 3-RPS parallel mechanism. The kinematics of the arm is analyzed using gradient projection method, and a redundant motion solving algorithm is proposed to construct a robot arm controller. An impedance control strategy combined with an improved Proportional-Integral-Derivative (PID) controller is adopted to realize soft contact with the wall surface. The research aims to speed up the wall treatment and shorten the project cycle through the interior wall finishing design model, providing a scientific theoretical basis for improving the technical level of the construction industry. The key contribution of the research lies in designing a composite robot system that integrates macro and micro robotic arms. This system can not only perform basic operations such as wall finishing, but also has adaptive functions to cope with non-standard and complex wall environments. By introducing an active compliance control algorithm and precisely adjusting the contact force and motion state, the adaptability between the robot and the wall surface is ensured, thereby enhancing the quality and safety of finishing operations.

The research content is mainly divided into four sections. The first section is a literature review, which examines the current research status and existing deficiencies of wall decoration robots. The second section elaborates on the construction method of motion redundancy optimization and active compliance control algorithms for interior wall decoration design models. The third section verifies the validity and practicability of the proposed model through experimental analysis. Finally, the fourth section summarizes the research conclusions and proposes the directions for future research. By systematically elaborating on these contents, readers can gain a comprehensive understanding of the background, methods, and potential applications of this study in the architectural decoration industry.

Literature review

Since the birth of robotics technology, its continuous progress has made many research institutions and companies realize the enormous potential of introducing it into construction. For example, Wang et al. proposed an interactive, immersive process-level digital twin for virtual reality environments. The system incorporated visualization and monitoring, task planning and execution, and interactive communication with a robot. The main responsibilities of robots included sensing and monitoring the work area, precise motion planning, and actual work. After a brief training, users could effectively organize job steps, select the best task plan, and guide the robot’s trajectory.⁶ Garrido et al. designed a new wall-climbing inspection robot, which utilized adsorption forces generated by permanent magnets to attach to rebar to facilitate safety-critical concrete structures for climbing. To ensure that the robot operates efficiently on re-bars with different arrangement patterns and on surfaces covered with a protective layer of 30 to 35 mm concrete, simulations and field tests were conducted to determine the optimal configuration of flux-focused permanent magnets.⁷ Kim et al. innovatively combined building information modeling techniques with robotic task planning to generate a detailed for executing robot action sequences. The research revealed how BIM technology could be utilized to integrate the fields of architecture and robotics to plan and guide automated robotic operations in construction projects.⁸ Liang et al. proposed a new classification method for collaborative human-robot work in construction teams. The classification was based on the robot’s ability to operate independently and the corresponding amount of work required by the human team members. The classification method provided new research paths and ideas for subsequent research.⁹

Kulkarni et al. designed a highly automated robotic device specifically constructed to perform wall painting operations. When the robot was activated, it utilized an in-built vertical movement system to carefully paint in the vertical direction starting from the bottom of the wall. Once painting was completed in a vertical area, the robot moved laterally to the next untreated area and continued to perform the painting job. Repeating this process until all walls were evenly painted. The entire wall could be automatically painted.¹⁰ Hartmann et al. proposed a new planning system that parallelized complex tasks and motion planning by decomposing them into multiple small-scale sub-problems. The system combined optimization techniques and sampling-based bi-directional spatio-temporal path planning techniques to efficiently handle multi-robot cooperative operations with uncertain arrival times. The planning system demonstrated excellent robustness in long-term planning and scalability in handling multiple objects and agents.¹¹ Cai et al. proposed a novel robot path planning strategy that combined deep reinforcement learning with construction worker motion prediction to promote efficient and safe human-machine collaboration in construction scenarios. The research results showed that the robot reached the specified destination with a 100% success rate in a near shortest path.¹² Yu et al. developed an integrated indicator to characterize the receiver’s grip state, including grip strength, posture, and full tactile sensing of the hand. The system was designed to simulate human grip movements and apply it to robotics. It was experimentally validated that this method was superior to techniques that relied on a single perspective to simulate the natural handover of objects during the transfer process from robots to humans, thereby improving the safety of workers when transferring materials at close range.¹³

In summary, the research and application of decorative robots mostly focus on specific processes with a single function. The working environment is usually a constantly changing non-standardized environment, but most of the existing robots are tested under ideal conditions in the laboratory, and there are adaptability problems when facing the complex real environment. Most remodeling robots still require human involvement to operate, and are also not sufficient to replace humans in delicate operations. The construction quality of robots cannot catch up with manual construction, and usually can only complete tasks with lower quality requirements. To address the above problems, a design model for interior wall decoration with motion redundancy optimization and active compliance control algorithm is proposed.

Model design of interior wall finishing design incorporating motion redundancy optimization and active supple control algorithm

Design of indoor wall finishing robotic arm based on motion redundancy control algorithm

Macro-motion robots usually refer to robots with a large range of motion and strong power output. They can perform large-scale, high-speed movements and can be used for industrial production, logistics and handling, construction, and more. By precise control and powerful load capacity, various complex tasks can be completed to improve production efficiency and quality.¹⁴ Robotic arms need to meet deadweight and load requirements, and have sufficient travel, compact shape, and sensitive response. Based on these functional requirements, the structure of the micro robotic arm adopts a direct drive parallel mechanism, which is configured as a 3-RPS parallel mechanism, as shown in Figure 1.¹⁵

Figure 1.

Structure and coordinate system of 3-RPS parallel mechanism.

Figure 1 illustrates the structure and coordinate system of the 3-RPS type parallel mechanism. The center point Og of the moving platform has linear and angular velocities, the rod has directional unit vectors, and the branching drives the rod to move at speed i. The parts are connected by ball hinges and their positions have corresponding velocities in a fixed coordinate system. The mechanism is chosen as a form of micro robotic arm structure, because it can meet the similar wrist adjustment attitude similar to that of the micro robotic arm in wall operations. 3-RPS type parallel mechanism has two rotational degrees of freedom, which facilitates the fast and accurate attitude adjustment. Meanwhile, it meets the functional requirements on self-weight and load, travel, compactness, and response speed. The matrix is shown in equation (1).

{}_{B}^{A}R = [\begin{array}{c} \cos β \cos y & - \cos β \sin y & \sin β \\ \sin α \sin β \cos y & - \sin α \sin β c y + \cos β \cos y & - \sin β \cos y \\ - \cos α \sin β \cos y + \sin α \sin y & \cos α s β \cos y + \sin α \sin y & \cos α \cos β \end{array}]

(1)

The expression for the fixed coordinate system ${A}$ is given in equation (2).

{}^{A}L_{i} = {}^{A}S_{i} - {}^{A}R_{i} = {}_{B}^{A}R_{i} {}^{B}S_{i} + {}^{A}P_{B O} - {}^{A}R_{i}

(2)

In equation (2),

L_{i}

denotes the fractional actuator.

R_{i}

represents the rotating sub-hinge point on

{A}

S_{i}

represents the ball hinge sub-hinge point on

S_{1} S_{2} S_{3}

. The kinematic inverse solution of the 3-RPS parallel mechanism can be obtained by performing vector modulo operation on equation (2). The rotating sub-hinge points in the 3-RPS mechanism are arranged in such a way that the ball hinge points connected to them always move along the plane perpendicular to the stationary platform. From the ball-hinge point coordinate

S_{i}

, the attitude constraint equation shown in equation (3) is derived.

{\begin{cases} {}^{A}X_{B O} = \frac{1}{2} r_{b} (\cos β \cos γ + \sin α \sin β \sin γ - \cos α \cos γ) \\ {}^{A}Y_{B O} = r_{b} \cos β csin γ \\ γ = - r_{a} \frac{\sin α \sin β}{\cos α \cos β} \end{cases}

(3)

For a 3-RPS mechanism, its workspace is the set of attitude angles that can be achieved by translating the moving platform along the Z-axis. For the 3-RPS mechanism, its workspace is solved with the help of MATLAB programming and the optimization design is carried out.^16–18 The distribution of the joint coordinate system of the macro robotic arm is shown in Figure 2.

Figure 2.

Distribution of joint coordinate system of macro robotic arm.

It is assumed that the Jacobi matrix of the macro/micro robotic arm as a whole is shown in equation (4).^19,20

J_{M - m} = [J_{M} J_{m}]

(4)

In equation (4),

J_{M}, J_{m}

denote the partial velocity mapping matrices of the macro robotic arm and micro robotic arm, respectively. Since the macro robotic arm is a tandem structure, its partial velocity mapping matrix is solved by the geometric method, which is shown in equation (5).²¹

J_{M} (i) = [\begin{array}{c} z_{i} \times r_{i} \\ z_{i} \end{array}] = [\begin{array}{c} z_{i} \times ({}^{W}P_{t c p} - {}^{W}P_{i}) \\ z_{i} \end{array}]

(5)

In equation (5),

z_{i}

represents a unit vector that rotates along the hinge coordinate system

i

of the robotic arm.

r_{i}

is the position vector of the origin of the macro robotic arm terminal coordinate system under the joint coordinate system

i

{}^{W}P_{t c p}

represents the position vector of the center point of the end effector of the macro/micro robotic arm in the world coordinate system.

{}^{W}P_{i}

is the position vector of the origin of the hinge

i

coordinate system of the macro robotic arm in the global coordinate system. The velocity mapping matrix of the micro robotic arm part is shown in equation (6).

J_{m} = [\begin{array}{l} {}_{5}^{W}R & 0 \\ 0 & {}_{5}^{W}R \end{array}] J_{R P S}

(6)

In equation (6),

J_{R P S}

denotes the Jacobi matrix.

{}_{5}^{W}R

denotes the rotation transformation matrix of the coordinate system of joint 5 on the macro robotic arm relative to the world coordinate system. The generalized Jacobi matrix can be obtained from equation (5) and (6). The kinematic velocity of the robotic arm is shown in equation (7).

\dot{X} = J \dot{θ}

(7)

In equation (7),

\dot{θ}

denotes the joint velocity of the robotic arm. When obtaining the weight matrix, equation (8) is obtained.

\dot{θ} = J^{+} \dot{X} + (I - J^{+} J) \dot{φ}

(8)

In equation (8),

I

denotes the unit matrix.

J^{+}

denotes the generalized inverse of

J

, which defines the motion of the end of the machine and produces the end velocity

\dot{X}

\dot{φ}

represents an arbitrary vector. Gradient projection method is an optimization method used to solve nonlinear planning problems with constraints. The gradient projection method is to find a descent direction in the feasible domain through gradient projection technique, so as to reduce the objective function value while satisfying the constraint requirements. Its computational expression is shown in equation (9).²²

\dot{φ} = K \cdot \nabla H

(9)

In equation (9),

K

is a scalar coefficient. The key element of the free vector is

\nabla H

, the chi-square solution of equation (8) can be used to minimize a specific quantity

H (θ)

at a high level without interfering with the end motion of the robot. The performance function is set as equation (10).

{\begin{cases} H (θ) = \frac{1}{n} \sum_{i = 1}^{n} (\frac{θ_{i} - α_{i}}{α_{i} - θ_{i \max}}) \\ α_{i} = (θ_{i \max} + θ_{i \min}) / 2 \end{cases}

(10)

The joint optimization is to minimize the objective value in equation (10). Considering the property that the joints of the micro robotic arm have limited motion, the arm will select different $K$ values. In this case, the expression for the free vector $\dot{φ}$ follows equation (11).

\dot{φ} = {\begin{cases} K_{M} \sum_{i = 1}^{9} (\frac{θ_{i} - α_{i}}{α_{i} - θ_{i \max}}) (m a c r o m a n i p u l a t o r) \\ K_{m} \sum_{i = 1}^{3} (\frac{θ_{i} - α_{i}}{α_{i} - θ_{i \max}}) (m i c r o m a n i p u l a t o r) \end{cases}

(11)

In equation (11), the amplification factors corresponding to the macro/micro robotic arm sections are

K_{M} = 0

and

K_{m} = 1 - 2.025 L^{2} \times (3 - 2 L)

, respectively. The operation steps of the redundant motion simulator for the macro/micro robotic arm of the decoration robot are shown in Figure 3.

Figure 3.

Redundant motion simulator for macro/micro robotic arm of decoration robot.

Construction of interior wall finishing model incorporating active active compliance control algorithm

There are two implementation methods for robot active compliance control. One is passive compliance control, and the other is active compliance control.²³ Active pliable control focuses on the force-position relationship, which is categorized into four strategies: impedance control, force/position hybrid, adaptive control, and intelligent control. The latter two are based on the combination of traditional force control theory and advanced control theory. Electrical impedance describes the ratio of voltage to current, and robot impedance control follows this ratio to regulate the ratio of contact force to position for detailed control. The robot can be simplified as a single-degree-of-freedom system. The impedance is expressed by the second-order dynamics equations of the $m$ mass- $c$ damped-spring model, as shown in equation (12).

m \ddot{x} + c \ddot{x} + k x = f

(12)

In equation (12),

k

denotes the stiffness coefficient.

f

is the environmental contact force. To track accurately, the study introduces a target contact force

F_{d}

to differ from the actual contact force

F

and substitutes it into equation (13) to obtain a new target impedance model.

{\begin{cases} M_{d} \ddot{X} + B_{d} \dot{X} + K_{d} (X - X_{d}) = F_{e} \\ M_{d} \ddot{X} + B_{d} (\dot{X} - {\dot{X}}_{d}) + K_{d} (X - X_{d}) = F_{e} \\ M_{d} (\ddot{X} - {\ddot{X}}_{d}) + B_{d} (\dot{X} - {\dot{X}}_{d}) + K_{d} (X - X_{d}) = F_{e} \end{cases}

(13)

In equation (13),

F_{e}

denotes the contact force deviation. To achieve more accurate control force, the 3rd target impedance model is usually chosen. Between the robot and the environment, a module labeled “impedance” is used to represent the impedance characteristics of the robot. In contrast to the impedance module, a module labeled “conductance” represents the conductance characteristics of the robot. The model is shown in Figure 4.

Figure 4.

Relation model of impedance and admittance between robot and environment. (a) Impedance relation model (b) Admittance relation model.

In Figure 4(a), $Z$ represents the system impedance. The impedance relationship is triggered when there is a deviation between the actual position of the robot end $X$ and the target position $X_{d}$ . In this case, the impedance model of the robot comes into play. Based on the deviation, the model drives the robot end to generate the corresponding output force $F_{e x t}$ . The impedance control technology of force essentially achieves the adjustment of motion contact force and motion state by converting the changes in the end effector motion trajectory into joint torque, characterized by precise modeling of the motion model and torque control. In Figure 4(b), $1 / H$ denotes the system conduction model. In the process of interaction between the robot and the environment, there exists a conductivity relationship, which is specifically manifested as follows. Once the robot end is subjected to the force from the environment $F_{e x t}$ , the position of the robot end will correspondingly appear the deviation $E$ . The active compliance actuator used for exterior wall finishing is a complex electromechanical system, which has the functions of a putty scraper tool, as well as contact force detection and positioning adjustment. The compliance control strategy flow of the finishing actuator system is shown in Figure 5.

Figure 5.

Flow chart of pliable control strategy of finishing actuator system.

The compliance control strategy flow of the finishing actuator system shown in Figure 5 focuses on collecting data from force sensors to achieve soft contact between the system and the wall. During the finishing operation, the force sensor collects the contact force data between the finishing actuator and the wall surface in real time and inputs it to the system controller. After receiving the data, the system controller is processed by internal arithmetic and outputs an angular displacement instruction. This instruction is used to drive the actuator action so that the actuator can make adaptive adjustment according to the actual situation of the wall surface and realize the soft contact with the wall surface. In this process, the system selects the active compliance control system composed of impedance controller and improved PID controller. The improved PID control introduces the integral separation mechanism to suppress the oscillation in the steady state phase. With such a control strategy, the system can precisely regulate the contact state between the finishing actuator and the wall surface to ensure stable and precise finishing operations under different wall conditions and improve the quality and efficiency of finishing. The control rate of the conventional incremental PID controller is shown in equation (14).

Δ μ = k_{p} (e_{n} - e_{n - 1}) + k_{i} e_{n} + k_{d} (e_{n} - 2 e_{n - 1} + e_{n - 2})

(14)

In equation (14),

k_{p}, k_{i}, k_{d}

represent the coefficients of proportional, integral and differential links, respectively. The above controller is prone to cause significant jitter in the convergence stage of the contact force near the steady state. To solve the problem, the study introduces the integral separation mechanism to improve the traditional PID control strategy and suppresses the oscillation phenomenon in the steady state stage by dynamically adjusting the strength of the integral term during the contact force regulation process, as shown in equation (15).

{\begin{cases} Δ μ = k_{p} (e_{n} - e_{n - 1}) + k_{i} e_{n} + k_{d} (e_{n} - 2 e_{n - 1} + e_{n - 2}) \\ β = {\begin{cases} 1, | e_{n} | \leq ε \\ 0, | e_{n} | \leq ε \end{cases} \end{cases}

(15)

In equation (15),

β

is the integral separation coefficient, which is used to dynamically adjust the intensity of the integral term in the control output.

ε

represents the error threshold.

e_{n}

represents the error at the current moment. The control rate output is shown in equation (16).

μ_{n} = μ_{n - 1} + Δ μ

(16)

The basic principle of impedance control for plaster actuator in the study is shown in equation (17).

f_{i m} = K_{i m} (θ_{d e s} - θ_{a c t}) + D_{i m} ({\dot{θ}}_{d e s} - {\dot{θ}}_{a c t})

(17)

In equation (17),

K_{i m}, D_{i m}

denote the desired system stiffness and damping, respectively.

f_{i m}

denotes the impedance system output force/moment.

θ_{d e s}, {\dot{θ}}_{d e s}

denote the desired angle and desired velocity, respectively.

θ_{a c t}, {\dot{θ}}_{a c t}

denote the actual angle and actual speed, respectively. The error amount is expressed in equation (18).

e_{n} = f_{a c t} - f_{i m}

(18)

The control output calculation process is shown in equation (19).

u = u_{n} + K_{f I} f_{i m}

(19)

In equation (19),

K_{f I}

denotes the motor torque/current conversion factor. Figure 6 shows the schematic diagram of the interior wall trim design model incorporating motion redundancy optimization and active supple control algorithm.

Figure 6.

Macro/micro robotic arm decoration structure.

In Figure 6, this interior wall finishing model is divided into four main parts, that is, mobile platform, macro robotic arm, micro robotic arm, and finishing actuator. The mobile platform is configured with a height-adjustable system, which enables the mounted robotic arm to perform high level operations. The arm used is a lightweight UR5 six-degree-of-freedom collaborative robotic arm with a weight capacity of 5 kg, an arm span of 0.85 m, and a joint rotation speed of up to +180 9% s, which makes it suitable for finishing operations that require flexibility in spatial movement. In addition, a small parallel arm with three-degree-of-freedom is developed, capable of lifting at least 2 kg and incorporated with a laser distance sensor to accurately control the position of the finishing scraper. The finishing actuator consists of a servomotor, a speed reducer, a torque sensor, and a stainless steel scraper.

Experimental analysis of the wall finishing model of the decoration robot

Simulation performance verification of decorative robots

The study is based on wall plaster control to analyze the effect of wall finishing by the renovation robot, and the experiment is carried out in a constant temperature and humidity laboratory (23 ± 1°C, humidity 45% RH). The wall simulation platform consists of 12 mm thick calcium silicate boards (2400 × 1200 mm) vertically mounted on a steel frame, with the surface flatness calibrated to ±0.3 mm. The UR5 six-degree-of-freedom collaborative robotic arm is used as the main operating unit, with a repeatable positioning accuracy of ±0.1 mm when the end load is 5 kg, and the range of joint motion is limited to ±90°. The micro-adjustment mechanism uses a 3-RPS parallel configuration with a Maxon EC-i40 hollow-cup motor, and realizes 125-150 mm stroke adjustment through a T-screw. The finishing actuator integrates Denso six-dimensional torque sensor and KEYENCE laser distance measuring module. The scraper is made of 304 stainless steel, which realizes normal compensation through THK linear module, with dynamic positioning accuracy of ±0.01 mm. The control system is based on NI cRIO-9049 real-time controller, and the motion planning is based on gradient projection algorithm, with the impedance controller parameterized as M_d = diag (2,2,2) kg-m², B_d = diag (15,15,15) N-s/m, and K_d = 10 N/m. In the experiment, the scraper travels at a speed of 80 mm/s, the wall contact force is controlled at 15 ± 2 N, and the moving platform lifts with a motorized actuator, and the data are collected by LabVIEW with a 2 ms cycle. The D-H parameters of macro robotic arm can be obtained according to the D-H parameter method, as shown in Table 1.

Table 1.

D-H parameters of macro robotic arm.

Articulation	1	2	3	4	5	6
A_a-1 (mm)	0	0	−425	−392.4	0	0
B_a-1 (rad)	0	π/2	0	0	π/2	−π/2
C_a-1 (mm)	89.4	0	0	109.1	94.6	82.3
D_a-1 (rad)	0	0	0	0	0	0
Joint range (rad)	−π/2∼π/2	−π/2∼π/2	−π/2∼π/2	−π/2∼π/2	−π/2∼π/2	−π/2∼π/2

In Table 1, there are several key variables to be clarified when describing the kinematic parameters of the robot arm. i denotes the i-th joint of the robotic arm. A_a-1 and B_a-1 correspond to the spatial distance and rotation angle between the Z-axis of the coordinate system of joint i-1 and the Z-axis of the coordinate system of joint i, respectively. C_a-1 refers to the distance between the origin of the coordinate system of joint i-1 and the origin of the coordinate system of joint i in the direction of Zi-1 axis. As for D_a-1, it is the rotation angle of joint i-1 around the Z-axis of its own coordinate system. The study is calculated by MATLAB theory to obtain the macro/micro robotic arm trajectory tracking position and tracking attitude accuracy results.

From Figure 7(a), the calculated results of the trajectory tracking position of the macro/micro robotic arm in the X, Y, and Z-axis had almost no deviation from the actual position. From Figure 7(b), the actual and calculated values of the trajectory tracking attitude of the macro/micro robotic arm had no deviation in the α-Euler angle and γ-Euler angle. Only at the β-Euler angle, there was a deviation of approximately 0.1° between the actual trajectory tracking attitude value and the computer value displayed. In summary, the trajectory tracking accuracy of the macro/micro robotic arm is within the allowable error range. Figure 8 shows the redundant motion simulation results of the macro/micro robotic arm.

Figure 7.

Macro/micro robotic arm trajectory tracking position and attitude accuracy. (a) Macro/micromanipulator arm trajectory tracking position accuracy (b) Macro/micromanipulator arm tracking attitude accuracy.

Figure 8.

Simulation results of redundant motion of macro/micro robotic arm. (a) The length of the driving rod of the micromanipulator arm/ the change of the joint Angle of the macro manipulator arm (b) Micromanipulator drive speed.

Figure 8(a) shows the relationship between the length of the driving rod of the micromanipulator arm and the joint angle of the macro robotic arm over time. The length of the driving rod of the micro robotic arm fluctuated between 0.125 m and 0.150 m over time, while the joint angle of the macro robotic arm showed a decreasing trend. Figure 8(b) depicts the drive velocity components of the micro robotic arm, including the v component and the ω component. v component fluctuated between −30 and 30, while the ω component fluctuated between −15 and 15. These data indicate that the joint motion and drive speed are within reasonable ranges, and effective limiting and speed limiting are realized. By coordinating the macro robotic arm, the system is able to move in a wide area of space without affecting the precise attitude control of the robotic arm’s end effector. The micro robotic arm is able to synchronize the attitude adjustment when the macro robotic arm moves horizontally along the wall, thus realizing complex motion decomposition and obstacle avoidance adjustment, and ensuring the efficiency and accuracy of the overall system. Figure 9 shows the results of the position response to the external contact force under different K values.

Figure 9.

Response of positions to external contact forces under different K values. (a) K=1 (b) K=2 (c) K=50.

As shown in Figure 9(a) and (b), the flexibility of the system could be effectively changed by adjusting the stiffness coefficient under a consistent external force. In the plaster actuator system, the flexibility was inversely proportional to the stiffness coefficient, that is, the lower the stiffness coefficient, the higher the flexibility of the system. According to Figure 9(c), when K = 50, there was a clear oscillation in the position curve. This oscillation persists on the time axis, indicating that the system is unable to maintain stability even after it enters the steady state, which is a typical manifestation of too high stiffness coefficient leading to excessive rigidity of the system and difficulty in adapting to changes in external forces. Too high stiffness coefficient will destroy the force-position synergistic relationship, so that the positional response cannot precisely follow the change of the external force, which weakens the stability of the system. The tracking results of the system to the external force under different k values are shown in Figure 10.

Figure 10.

The system’s tracking of external forces under different k values. (a) K=1 (b) K=2 (c) K=10 (d) K=50.

In Figure 10, the force tracking performance of the system varied with the variation of the stiffness parameter k. In Figure 10(a) and (b), when k was 1 and 2, the measured force and the desired force curves were basically coincident between 0 and 0.5 s, with a slight offset. When k = 1, the measured force was slightly lower than the desired force near 0.2 s. However, the overall trend was consistent, which indicated that the system was able to follow the change of external force better. From Figure 10(c) and (d), at k = 10 and 50, although the measured force was still close to the desired force in the initial stage, the measured force curve showed obvious up and down jitter with time, which deviated from the desired force. For example, at k = 50, the jitter was intensified after 0.4 s. This indicates that the system can follow the change of external force better. The stiffness coefficient should be no more than 10 under the premise of ensuring accurate force and position following and system stability. Excessive stiffness can cause the system to become too rigid, resulting in frequent small oscillations and instability when subjected to force. Figure 11 shows the force and position during finishing with different robotic arms.

Figure 11.

Experimental results of interior wall decoration design model. (a) Comparison of the end tracks of the two robot arms moving along the wall in a straight line (b) The change of force and position during the contact between the plaster actuator and the wall.

In Figure 11(a), the end trajectory of the six-degree-of-freedom robotic arm nearly fitted an ideal straight line, with the y-coordinate decreasing steadily with x, compared with the steep decline in the back section of the nine-degree-of-freedom robotic arm trajectory. The contrast shows that the algorithm has a good layout in terms of degree-of-freedom management rather than just stacking degree-of-freedom. In Figure 11(b), the distance between the wall and the moment remained stable in the initial stage. After 10 s, there was a strong contrast between the sudden increase in moment and the sudden decrease in wall distance, but the overall fluctuation trend was still within the controllable threshold. This verifies that the controller continuously calibrates actions between force and pose, anchoring key targets for force control optimization in subsequent intelligent construction operations. The research aims to quantify the comparison of the trajectories of six-degree-of-freedom and nine-degree-of-freedom robotic arms. The results are shown in Table 2.

Table 2.

Quantitative analysis results of the comparison between six-degree-of-freedom and nine-degree-of-freedom robotic arms.

Indicator	Project	Six-degree-of-freedom robotic arm	Nine-degree-of-freedom robotic arm
Position error (mm)	Straight line segment	0.82	0.05
	Corner section	2.35	0.25
	Overall RMS	1.48	0.11
Attitude angle error (°)	α (Roll)	1.32	0.08
	β (pitch)	1.45	0.1
	γ (yaw)	0.95	0.07
Path following property	Maximum overshoot	3.2 mm	0.18 mm
Path following property	Convergence time	0.65s	0.12s

Table 2 compares the quantitative analysis results of the trajectory tracking performance of six-degree-of-freedom and nine-degree-of-freedom robotic arms. The displacement errors of the six-degree-of-freedom robotic arm in the straight line section and the corner section were 0.82 mm and 2.35 mm, respectively, while the nine-degree-of-freedom robotic arm was significantly superior, being 0.05 mm and 0.25 mm, respectively. From the perspective of overall root mean square error, the six-degree-of-freedom robotic arm was 1.48 mm, while that of the nine-degree-of-freedom robotic arm was only 0.11 mm, highlighting the significant advantage of the latter in terms of accuracy. In terms of attitude angle errors, the six-degree-of-freedom robotic arm had relatively high errors in pitch, roll, and yaw angles, which were 1.32°, 1.45° and 0.95°, respectively, while the errors of the nine-degree-of-freedom robotic arm were significantly reduced to 0.08°, 0.1° and 0.07°. In terms of path tracking characteristics, the maximum overwear and convergence time of the nine-degree-of-freedom robotic arm were significantly better than those of the six-degree-of-freedom robotic arm, which were 0.18 mm and 0.12s, respectively. These results indicate that the nine-degree-of-freedom robotic arm demonstrates significant advantages in both accuracy and response speed, making it suitable for application scenarios that require higher flexibility and precision.

Real-world deployment verification of decorative robots

To verify the parameter sensitivity of the decorative robot in the real world, the study analyzes the performance of the wall decoration robot under different stiffness (K) and damping (D) parameter settings. The influence of these parameters on the stability, accuracy and response speed of the finishing actuator system is evaluated. The stiffness (K) and damping (D) parameters of the robot finishing actuator are adjusted, and four sets of parameter combinations are set, respectively. Option One: K = 1 and D = 5; Option Two: K = 10 and D = 10; Option Three: K = 20 and D = 15; Option Four: K = 30 and D = 20. The experimental results under different combinations of stiffness (K) and damping (D) are shown in Table 3.

Table 3.

Experimental results under different combinations of stiffness (K) and damping (D).

Project	Option one	Option two	Option three	Option four
Stiffness (K)	1	10	20	30
Damping (D)	5	10	15	20
Average finishing quality (mm)	0.15	0.1	0.08	0.12
Oscillation frequency (Hz)	4.5	3.0	2.0	2.5
Deviation (N) of contact force with the target	2.3	1.5	1.0	2.0
Average response time (ms)	150	120	100	90

Table 3 presents the experimental results of the wall decoration robot under different combinations of stiffness and damping parameters. With the increase of stiffness and damping, the finishing quality gradually improved from Option One to Option Three. The average finishing quality of Option Three, with the best effect, at 0.08 mm. The oscillation frequency of Option One was the highest, reaching 4.5 Hz. However, as the parameters increased, the oscillation frequency dropped to the lowest, at 2.0 Hz, indicating a significant improvement in the system stability. Furthermore, the deviation of the contact force with the target was the lowest in Option Three, only at 1.0 N, demonstrating enhanced accuracy. The average response time also decreased with the increase of stiffness and damping. The response time of Option Four was 90 ms, which was the best performance. To quantify the contribution of each subsystem to the overall system performance, the study analyzes the position error, β angle error, contact force fluctuation, and mortar thickness variation of each subsystem. The specific results are shown in Table 4.

Table 4.

The contribution results of each subsystem to the overall system performance.

Subsystem status	Positional error contribution (mm)	β angle error contribution (°)	Contact force fluctuation contribution (N)	Variation in mortar thickness (mm)
Noise from the motion planning module	0.07	0.01	0.2	0.02
Macro arm execution (increased friction	0.04	0.02	0.7	0.06
Use only macro arms (no micro arms)	0.23	0.13	2.1	0.26
Sensor noise interference	0.02	0.02	1.4	0.11
No active compliance control	0.35	0.16	4.4	0.46
Environmental disturbance (3 mm protrusion	0.06	0.04	1.2	0.09

Table 4 shows the contribution of each subsystem to the overall performance of the wall decoration robot. The influence of different factors on the system performance varied significantly. The lack of active compliance control had the greatest impact on system performance, resulting in significant increases in position error, β angle error, contact force fluctuation, and mortar thickness variation, demonstrating the crucial role of this subsystem in control accuracy. When only macro arms were used without micro robotic arms, the errors and fluctuations of the system also increased significantly, indicating the importance of micro control arms in handling details and complex structures. Meanwhile, the increase in friction and the noise from the motion planning module also had a certain impact on the dimensional accuracy and the stability of the contact force. The research designed a detailed real-world deployment verification experimental plan for the robot model of interior wall decoration. This plan aims to evaluate the robustness of the system under different wall materials and geometries, and conducts a comparative analysis with existing methods. Wall materials include cement walls, brick walls, and lightweight partition walls. Geometric shapes include entire walls, walls with door and window openings, and curved walls. The specific comparison methods are traditional manual finishing, MEYER robots, and robots that only use macro robotic arms and rigid control. The specific results are shown in Table 5.

Table 5.

Results of real-world deployment verification experiments.

System	Research method	Commercial MEYER robot	Only macro arm + rigid control	Manual finishing
Position error (mm)	0.12	0.25	1.8	/
Attitude angle deviation (°)	0.05	0.11	0.75	/
Force tracking error (N)	0.8	1.2	4.5	1.0
Force oscillation amplitude (N)	2.1	3.5	12.0	3.0
Stabilization time (s)	0.3	0.7	3.2	0.5
Thickness uniformity (mm)	0.15	0.2	0.85	0.3
Flatness (mm/2m)	0.4	0.6	2.5	1.0
Efficiency (m²/h)	1.2	1.8	3.5	2.0
Energy consumption (kWh/m²)	0.8	1.1	2.4	1.2

Table 5 presents the performance evaluation results of the indoor wall decoration robot model in the real-world deployment verification experiment. Compared with traditional manual finishing, MEYER robots and robots that only use macro robotic arms with rigid control, the research method showed certain advantages. In terms of position error and posture angle deviation, the research method had the highest accuracy, which were 0.12 mm and 0.05°, respectively, while the error of traditional manual finishing did not quantified. The research method was also far superior to other methods in terms of contact force tracking error and the oscillation amplitude, indicating its stronger stability. The stabilization time was 0.3 s, which was also lower than that of other comparison methods, indicating a more rapid response. In terms of thickness uniformity and wall flatness, the research method also performed outstandingly, with 0.15 mm and 0.4 mm/2m, respectively, outperforming the MEYER robot and the rigidly controlled robot. In terms of efficiency and energy consumption, although the efficiency of the research method was 1.2 m²/h, it still had an energy efficiency advantage on energy consumption of 2.4kWh/m². These results indicate that the research method has demonstrated excellent performance and feasibility when facing challenges from different wall materials and geometric shapes. It is expected to improve construction quality and efficiency in practical applications. The research assesses the influence of environmental factors such as temperature and humidity on the performance of interior wall decoration robots, and analyzes the performance results by comparing the robots of the research method with those of traditional manual finishing. The evaluation indicators construction time, cost, and quality. The research designs the laboratory environment as an adjustable temperature and humidity control space, setting three different environmental conditions. Condition one: Temperature 25°C, humidity 50%; Condition Two: Temperature 30°C, humidity 70%; Condition Three: Temperature 20°C, humidity 40%. Table 6 shows the performance results of the research method robot and traditional manual finishing under different temperature and humidity conditions.

Table 6.

Comparison of construction efficiency under different temperature and humidity conditions.

Environment condition	System	Construction time (min)	Cost (CNY/m²)	Thickness uniformity (mm)	Flatness (mm/2m)
25°C, 50%	Research method robot	15	30	0.15	0.4
25°C, 50%	Traditional manual finishing	25	40	0.30	1.0
30°C, 70%	Research method robot	18	32	0.12	0.5
30°C, 70%	Traditional manual finishing	30	42	0.35	1.2
20°C, 40%	Research method robot	12	28	0.18	0.3
20°C, 40%	Traditional manual finishing	28	38	0.25	0.8

Table 6 shows the construction efficiency of the indoor wall decoration robot and the traditional manual finishing under different temperature and humidity conditions. Under standard environmental conditions of 25°C and 50% humidity, the robot took 15 min to complete the finishing work, with a cost of 30 CNY/m². The thickness uniformity was 0.15 mm and the wall flatness was 0.4 mm per 2m. In contrast, traditional manual finishing took 25 min and costs 40 CNY/m². The thickness uniformity and the flatness of the wall surface were 0.30 mm and 1.0 mm, respectively, demonstrating the obvious advantages of the robot on time and quality. In a high-temperature and high-humidity environment of 30°C and 70% humidity, although the robot construction time slightly increased to 18 min and the cost was 32 CNY/m², compared with the 30 min and 42 CNY/m² of manual finishing, it still demonstrated higher efficiency and cost-effectiveness. In addition, under this condition, the thickness uniformity and wall flatness of the robot were also more outstanding, being 0.12 mm and 0.5 mm, respectively. In a low-temperature and low-humidity environment of 20°C and 40% humidity, the robot performed best, taking only 12 min and at a cost of 28 CNY/m². The thickness uniformity and wall flatness were 0.18 mm and 0.3 mm, respectively, which had significant advantages over manual finishing at 28 min, 38 CNY/m², 0.25 mm, and 0.8 mm. In the experimental results, the actual driving delay and hysteresis effect had a significant impact on the performance of the interior wall decoration robot. Although the control model assumes that the actuator can immediately respond to control instructions, in practical applications, the system often cannot achieve ideal instant feedback due to friction in the mechanical structure, response time of the actuator, and elastic deformation between connecting components. This delay may lead to fluctuations in contact force and unevenness in finishing thickness, thereby affecting the overall project quality. The robot in the research successfully reduces the negative impact in this regard through advanced motion planning and active compliance control strategies. For instance, in a high-temperature and high-humidity environment, although there is a certain lag effect during the experiment, the robot can still maintain excellent performance. The construction time and contact force errors are significantly lower than those of traditional manual finishing and robots that only use macro robotic arms. This indicates that with the help of multi-degree-of-freedom mechanical design and adaptive control, robots under the research method can timely correct their trajectories and adjust the applied forces with rich dynamic feedback.

Although the research focuses on the design and control of automated wall decoration robots, in the actual construction process, the ideal state of completely replacing human labor is often difficult to achieve. Therefore, the feasibility and effectiveness of human-machine collaboration are of great significance. Human–machine collaboration can adapt the construction process to various irregular and complex environments by combining the high-speed and precise operation of robots with the flexible decision-making of humans, thereby enhancing the overall construction efficiency. For instance, when it comes to complex wall designs and unexpected situations, humans can quickly adjust their strategies, while robots can maintain high efficiency in standardized tasks. Effective teamwork not only improves construction quality, but also reduces safety risks and enhances the adaptability of the working environment, thereby promoting the comprehensive intelligentization process of the industry.

Conclusion

To improve the quality of wall decoration, reduce labor intensity, and enhance the precision and efficiency of interior wall finishing, an innovative decorating robot finishing method was proposed. An intelligent decoration system integrating a macro/micro composite robotic arm and a force-controlled finishing terminal was designed, and a cooperative control architecture was established. The test results showed that the spatial error between the calculated value and the actual value of the end trajectory position of the robotic arm was lower than that of the detection threshold, and the attitude matching degree reached the standard of process requirements. The joint rotation angle of the mechanical body was limited to the preset safety interval, and the movement speed of the micro robotic arm was strictly in the preset safety range. In the force-position co-control of the finishing actuator, the optimal tracking stability was demonstrated when the system stiffness parameter was set to 10. With the help of a laser positioning system, the attitude parallelism error between the end of the robotic arm and the work plane is dynamically eliminated, confirming the effectiveness of redundant motion control logic. The study verified the effectiveness of redundant motion control algorithms for macro/micro robotic arms and active compliant controllers based on impedance control, further improving the parallel posture maintenance of the robotic arm end towards the wall and the adaptability of the finishing actuator to the wall. Although research has achieved certain results, there is still room for improvement. Future research will further optimize the robot’s ability to perceive and adapt to complex wall environments, and enhance its operational performance under different materials and textures. Meanwhile, the integration of active compliance control and force feedback mechanism will be explored in depth to better meet the diversified needs of wall finishing.

Footnotes

ORCID iD

Shijiao Han

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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