LITERATURE REVIEW OF FLEXIBLE ROBOT DYNAMIC
Applications of flexible robots
[80], [113], [127], [128], [130], [131] Applications of flexible robots can be seen in [34], [86], [91], [137], [138] The major applications of these robots are in space, medicine and nuclear technology
Figure 1.2 illustrates a flexible robot utilized in space technology, which significantly reduces energy consumption due to the limited number of robots deployed The workspace of these flexible robots can be expanded by increasing the length of their flexible links, while the control system remains less complex with fewer links involved An example is the Remote Robot System (RMS) used by NASA for various crucial tasks in space, operating at a low frequency of 0.04 to 0.35 Hz and an angular velocity of 0.5 degrees per second The RMS has a mass of 450 kg and can handle a tip load of 27,200 kg.
Flexible robots play a crucial role in microsurgery, particularly in challenging procedures within the confined spaces of the human body These advanced robotic systems assist surgeons in performing intricate operations, including neurosurgery, neck surgery, and heart surgery, which have historically posed significant difficulties due to their complexity.
Figure 1 2 The flexible robot in space
Figure 1 3 Flexible robot in medicine
Flexible robots play a crucial role in the nuclear field by performing essential and hazardous tasks, such as transporting and assembling radioactive rods They are designed to minimize radiation exposure while operating with low energy consumption and minimal interaction force with their environment, ensuring high precision and flexibility Additionally, these versatile robots find applications in various sectors, including military operations, machining, and construction.
Classifying joint types of flexible robots
The classification of flexible robots becomes easier based on determining the main types of joints used to design the robots
In a robotic system with n flexible links, each link i-1 and i is connected by a joint, which can be either a rotational joint or one of two types of translational joints (type P a or type P b) The kinematics of each flexible link i are influenced by the motion of the connecting joint and the elastic deformation of the preceding link i-1.
In systems where two links are connected by a rotational or translational joint P b, the motion of link i is influenced by the movement of joint i and the elastic deformation at the distal end of the preceding link i - 1 Conversely, in the case of a translational joint P a, the motion of link i is independent of the elastic deformation at the distal end of link i - 1, instead relying on the elastic deformation of the sliding element on link i - 1, which changes along its length over time.
In the analysis of rotational and translational joints, it is typically assumed that the elastic displacements at the first node of the initial element on link i are zero However, in the case of the translational joint Pb, the zero elastic deformation occurs at the sliding element of link I through the fixed translational joint It is crucial to consider the elastic effects of links when developing kinematic and dynamic models for flexible robots that incorporate all three joint types The variations in joint types lead to differences in solving the motion equations, highlighting the importance of understanding these distinctions in robotic modeling.
The single-link, two-link and multi-link flexible robots with only rotational joints are investigated in many studies for example [10], [12], [15], [24], [28],
[34], [37], [66], [72], [73], [88], [100], [103], [136], … There are some studies mentioning single-link flexible robots with translational joint P a or P b [13],
[23], [29], [73], [116], [133] However, combining types of joints in flexible robots is not yet fully and clearly considered in modeling and controlling.
Classifying flexible robots
The flexible robots are classified according to the number of joints and links, types of joints and their structures
1.3.1 The flexible robots with regard to number of links and joints
1 The single-link flexible robots
The single-link flexible robots are clearly investigated [13], [14], [26], [36],
[40], [48], [55], [64], [82], [101], [113], … The Fig 1.7 shows the single-link flexible robot with rotational joint and the Fig 1.8 describes the other with translational joint P b
Figure 1 7 The single-link flexible robot with rotational joint
Figure 1 8 The single-link flexible robot with translational joint
2 The two-link flexible robots The two-link flexible robots are studied in [15], [18], [23], [25], [29], [48],
Figure 1 9 The two-link flexible robots with only rotational joints
The study of two-link flexible robots primarily focuses on those equipped with rotational joints, as illustrated in Figure 1.9 Additionally, there are a few examples that incorporate both rotational and translational joints, as shown in Figure 1.10.
3 The planar serial multi-link flexible robots
The multi-link flexible robots (Fig 1.11) are studied in [10], [12], [15],
Figure 1 11 The planar serial multi-link flexible robots 1.3.2 Classifying the flexible robots according to structures
1 The series-link flexible robots
The flexible robots with series links are shown as Fig 1.11
2 The parallel-link flexible robots
The flexible robots with parallel links are described as Fig 1.12 [84]
Figure 1 12 The parallel-link flexible robots
Parallel robots are increasingly utilized across various applications, including entertainment, home services, aerial vehicles, submarines, and assembly robotics They offer significant advantages over serial robots, particularly in terms of handling heavy payloads and achieving high positional accuracy.
3 The mobile flexible robots [85] (Fig 1.13)
Figure 1 13 The mobile flexible robots
Flexible robots with mobile bases, including macro-micro robots, space robots, and underwater robotic vehicles, enhance operational efficiency in various tasks such as repair and maintenance, inspection, welding, cleaning, and machining These versatile machines extend the workspace, enabling more effective and precise execution of complex operations across diverse environments.
Modeling methods
In general, the flexible robots are the continuous systems characterized by
Therefore, these systems must be discretized into the finite elements to analyze the kinematic and dynamic
The AMM and FEM are mainly used in kinematic and dynamic modeling
The single-link and two-link flexible models with only rotational joints are usually described by AMM because of its efficiency [10], [18], [26], [27], [28],
[131], [136], [133], [135], [141], … There are many investigations using FEM in modeling the flexible robots [12], [15], [23], [24], [48], [64], [77], [103],
The authors in studies [39] and [80] demonstrate that the Finite Element Method (FEM) is more effective than the Analytical Multibody Method (AMM) for modeling flexible robots with combined rotational and translational joints, or those featuring multiple links with varying cross-sectional areas In FEM, each flexible link is subdivided into finite elements, allowing for the determination of kinetic and potential energy, as well as the calculation of mass and stiffness matrices based on joint variables and elastic displacements The number of elements in each link varies, resulting in different sizes for these matrices Consequently, constructing the global mass and stiffness matrices involves extensive calculations, complex transformations, and assembly, which poses significant challenges for flexible robots with intricate joint configurations These global matrices are essential for deriving the system's dynamic equations.
Differential motion equations
The differential motion equations can be described as
The AMM can be combined with Lagrange-Euler equations [10], [28], [37],
[47], [135] and with Gibbs-Appel equations [73], [100] However, these combinations are executed for single-link or two-link flexible robots with only rotational joints.
DYNAMIC MODELING OF THE PLANAR FLEXIBLE
Kinematic of the planar flexible robots
Let us consider the flexible planar robot consisting of n n ( Z ) links and n joints The arbitrary link i −1 is connected with a link i by a joint
The equation i i = n represents three types of joints: rotational joint (R) and translational joints P a and P b Each link i, with a length L i, is segmented into n i elements of equal length l ie Every element j of link i features two nodes, j and j + 1, where node j experiences flexural displacement.
(2 1) i j u − and a slope displacement u i (2 ) j Similarly, node j + 1 has a flexural displacement u i (2 j + 1) and slope displacement u i j (2 + 2)
Figure 2 1 A generalized schematic of an arbitrary pair of flexible links
The local coordinate system OXYiii is established for link i, with the origin Oi fixed at the proximal end of the link and the axis OXi pointing in the direction of link i Similarly, the coordinate system OXYi-1 is defined for link i-1 Additionally, OXY000 serves as the reference coordinate system fixed to the base.
H as the general homogeneous transformation matrix which transforms from the coordinate system O XY i i i to the coordinate system O X Y i − 1 i − 1 i − 1 This matrix is determined by executing in order of the below steps
Step 1 Translate the coordinate system O X Y i − 1 i − 1 i − 1 along i in the direction
O X − − to the position of joint i The homogeneous transformation matrix characterizing this translation is denoted as T( ) i This matrix is determined as
Step 2 Translate O X Y i − 1 i − 1 i − 1 , at the previous location, a long u ( 1) i − f in the direction O Y i − 1 i − 1 The homogeneous transformation matrix T(u ( 1) i − f ) describing this transformation is formulated as
Step 3 Rotate O X Y i − 1 i − 1 i − 1 , at the previous location, around Z i − 1 with rotational angle u ( 1) i − s The matrix homogeneous transformation R(u ( 1) i − s ) representing this rotation is defined as
Step 4 Rotate O X Y i − 1 i − 1 i − 1 , at the previous location, around Z i − 1 with rotational angle i The matrix homogeneous transformation R( ) i representing this rotation is described as cos sin 0 0 sin cos 0 0
It is note that from step 1 to step 4 as
If the joint i is the rotational joint, the parameters i ,u ( 1) i − f ,u ( 1) i − s , i are in turn the length L i − 1 , the flexural displacement
( 1)(2 i n i 1 2) u − − + at the end of link i −1 and the joint variable i
If the joint i is the translational joint P b , the parameters i ,u ( 1) i − f ,u ( 1) i − s , i are in turn the length L i − 1 , the flexural displacement
( 1)(2 i n i 1 2) u − − + at the end of link i −1 and the fixed angle i between link i −1 and link i
In the case where joint i is identified as the translational joint P a, the parameters i, u (1) i − f, u (1) i − s, and i correspond to the joint variable d i, the flexural displacement u (1)(2 i − k + 1), and the slope displacement u (1)(2 i − k + 2) at node k + 1 of element k This configuration represents the current position of joint i on link i − 1, along with the fixed angle i between link i − 1 and link i.
Step 5 Translate O X Y i − 1 i − 1 i − 1 , at the previous location, a long a i in the direction O X i i If the joint i is the rotation joint or translational joint P b , the value of a i is equal to zero If the joint i is the translational joint P a , the value of a i is d i −L i The general transformation matrix T ( ) a i of this step is established as
The general homogeneous transformation matrix which transforms from the coordinate system O X Y i − 1 i − 1 i − 1 to the coordinate system O XY i i i can be calculated as follows
( 1) cos sin 0 cos sin cos 0 sin
H , (2.7) where, the parameters i ,u ( 1) i − s ,u ( 1) i − f , , i a i are described in Tab 2.1
Table 2 1 The parameters i ,u ( 1) i − s ,u ( 1) i − f , , i a i depending on types of joints
Let us consider some specific cases If the joint i is the rotational joint, the matrix H i i− f ( 1) can be written as
If the joint i is the translational joint P a , the matrix H i i− f ( 1) is displayed as
If the joint i is the translational joint P b , the matrix H i i− f ( 1) is given as
( 1)(2 2) cos sin 0 ( )cos sin cos 0 ( )sin
The position vector of arbitrary point on the element j of link i in the coordinate system O XY i i i is determined as
( 1) ( , ) 0 1 T ij = j − l ie + x w x t ij r , (2.11) where x is the value of 0 x l ie and l ie is the length of the element j The elastic displacement w x t ij ( , ) at the point x is calculated as [12]
= , (2.12) where the shape functions m ( ),x m = 1 4 of the element j can be described as [12]
1 ( ) 1 3 2 ; ( ) 4 2 ; ( ) 2 2 3 2 ; ( ) 3 3 2 2 3 ie ie ie ie ie ie ie x x x x x x x x x x x x l l l l l l l
The position vector of arbitrary point on the element j of link i in coordinate system O X Y 0 0 0 can be found as
2.1.3 The kinematic relationship of 3 2 = 9 structures of the arbitrary two flexible links
The position vector of arbitrary point on all of links is determined based on the general homogeneous transformation matrix which is constructed above
There are 3 2 = 9 different structures of the arbitrary two flexible links when three types of joints (R, Pa, Pb) are combined
In the analysis of two-link flexible robots, we focus on the kinematic relationship where link i represents the second link When considering the links as rigid, the position vectors of the endpoints of link 1 and link 2 are denoted as r01 and r02, respectively However, when the links are flexible, these position vectors undergo changes that reflect the flexibility of the system.
The transformation of the coordinates is represented by T f = [fX fY] r, where r denotes the coordinates along the OX and OY axes The structures I (RR), II (RPa), and III (RPb) feature a rotational joint as their first joint, followed by R, Pa, and Pb joints as their second joints, respectively.
The robotic structures IV (PaR), V (PaPa), and VI (PaPb) feature the translational joint Pa, with the joint variable d1 representing the distance OO1 In contrast, structures VII (PbR), VIII (PbPa), and IX (PbPb) utilize the translational joint Pb as their first joint, where d1 indicates the distance from the first joint to the endpoint of link 1 Additionally, the second joint variable d2 is defined differently based on the type of joint: if joint 2 is a translational joint Pb, d2 measures the distance from joint 2 to the end-effector point; if joint 2 is a translational joint Pa, d2 corresponds to the distance O1O2 The kinematic relationship among these flexible links can be expressed through a total of nine unique structures.
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin sin
2 2 1(2 2) 2 2 2 cos sin 0 cos sin cos 0 sin
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
01_ 1 1 1 1 2 1 1 1 1 1 2 1 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
01 1 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
2 2 1(2 2) 2 2 2 cos sin 0 cos sin cos 0 sin
01 1 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
1 1 1 cos sin 0 cos sin cos 0 sin
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , , cos sin cos sin , sin cos sin cos
1 1 1 cos sin 0 cos sin cos 0 sin
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
1 1 1 cos sin 0 cos sin cos 0 sin
2 2 2 2 2 1(2 2) cos sin 0 cos sin cos 0 sin
01 1 1 1(2 1) 1 1 1 1(2 1) 1 cos sin 0 1 , cos sin sin cos 0 1
0 1 , cos sin cos sin , sin cos sin cos ,
DYNAMIC ANALYSIS AND POSITION CONTROL OF THE
Boundary conditions
The flexible robots without the translational joint P b are types I, II, IV and
V The first node of the first element of the flexible links is fixed in the rigid joint R or joint P a Therefore, we can assume that the elastic displacements at this first node are equal to zero It means u 11 = u 12 = 0 and u 21 = u 22 = 0 The size of matrices M,C K , is (2n 1 +2n 2 + 2) (2n 1 +2n 2 +2) because of the removal of rows and columns 2 nd , 3 rd ( u 11 = u 12 = 0) and (2 n 1 + 3) th , (2 n 1 + 4) th ( u 21 = u 22 = 0) of these matrices, respectively
3.1.2 The flexible robots with translational joint P b
Flexible robot types III, VI, VII, VIII, and IX feature a translational joint known as Pb, which causes their boundary conditions to vary over time Focusing on the two-link flexible robot of type VII, the first link is equipped with a Pb joint while the second link utilizes an R joint In this configuration, link 1 moves along the direction of the driving force Ft.
At time ti, element k of link 1 is positioned at joint Pb, as illustrated in Figure 3.1, without considering the length of the translational joint Node k is located within joint Pb, while node k+1 is situated outside of it We can assume that the elastic displacements at node k are zero, expressed as u12(k-1) = u12(k) = 0 The integer value of k can be determined accordingly.
At the next time step, the element adjacent to element k at joint Pb experiences zero elastic displacements, indicating that the boundary conditions are time-dependent This presents a significant challenge in solving the system's motion differential equations.
Following that, the rows and columns 2 k th , (2k +1) th of the matrices M,C, K are removed assuming u 1(2 k − 1) =u 1(2 ) k = 0 and combined the boundary conditions of link 2 with assuming u 21 = u 22 = 0 to become the matrices
The matrices M, C, and K are defined with dimensions (2n1 + 2n2 + ×2) and (2n1 + 2n2 + 2), respectively The vectors q undergo transformation into new vectors q* with a size of (2n1 + 2n2 + ×2) 1 It is important to highlight that the removal of rows and columns from these matrices and vectors occurs continuously over time, influenced by the changing value of k, which serves as a new variable in the motion differential equations of the system.
Forward dynamic
Analyzing the dynamic behaviors of system giving the forces/torques laws of the joints is the purpose of solving the forward dynamic of the system
The Runge-Kutta Nyström, Blaess-Gumpert, and fourth-order Runge-Kutta methods are commonly employed to solve the equations presented in Eq (3.2) using time domain discretization techniques for approximate results This dissertation utilizes the fourth-order Runge-Kutta method within MATLAB/SIMULINK software to effectively solve the motion differential equations of two-link flexible robots, owing to its widespread use and efficiency.
Specifically, let us consider the Eq (3.2) in the period T = t t 0 , f with the time step h The number of calculating steps is defined as
The initial values q q 0 , 0 of the first step time t i( =1) is given by
The vector f( , , ,t q q Q ex ) is the right side of the Eq (3.2) The approximate values of vector q(i +1), i = 1, , N − 1 following the time step
( 1) ( ) t i + =t i +h are determined thanks to the values of vector q( )i and the increment vectors k 1 ( ), ( ), ( ), ( )i k 2 i k 3 i k 4 i in time step t i( ) The steps can be described below as
+ The dynamic parameters of flexible robot ( , , , , n n L L m m 1 2 1 2 1 , 2 , )
+ The matrices and vectors M q ,C q q , K q Q( ) ( , ) ( ), ex ( )t
+ The initial parameters t t h N 0 , , , , , f q q 0 0 Step 2 Calculate the first step time t i( =1)
+ q(1) = q q 0 ; (1)= q 0 ; + q (1) = f ( (1), (1), (1), t q q Q ex (1)) Step 3 Set up the loop following the Runge-Kutta 4 th order algorithm
+ Calculate q(i+1), (q i+1), (q i+1) and save the calculated results
End Step 4 Give the results q , q , q( ) ( ) ( )t t t
The forward dynamic algorithm for flexible robots without the translational joint P b is illustrated in Fig 3.2, while the algorithm for flexible robots with the translational joint P b is presented in another figure.
Figure 3 2 The solving algorithm without the joint P b
Figure 3 3 The solving algorithm with the joint P
3.2.2 The results of numerical calculations
1 The flexible robot type I (RR)
In this section, the dynamic behaviors of flexible robot type I (RR) (Fig
The analysis of dynamic behaviors in flexible robots is conducted by examining the variation of payload through the forward dynamic problem Each flexible link is treated as an individual element, with key insights derived from the joint variables and the elastic displacement of the end-effector Figure 3.4 illustrates the schematic representation of the forward dynamic solution using the SIMULINK toolbox, where the driving torques at the rotational joints serve as input parameters The torque vector is defined by Equation (2.39), while Table 3.1 outlines the dynamic parameters specific to flexible robot type I.
Figure 3 4 The schematic of the solving forward dynamic on SIMULINK Table 3.1 The dynamic parameters of flexible robot type I
The dimensions of link 1 and link 2 are as follows: the lengths (L1 and L2) measure 1 meter and 0.5 meters, respectively Both links have a width of 0.02 meters and a thickness of 0.003 meters Additionally, the inertial moments for motors 1 and 2 are both calculated to be 5.86 x 10^-4 kg.m².
Table 3 1 The dynamic parameters of flexible robot type I (continuous)
The mass of the motor 2 mdc2 (kg) 0.155
The mass density of the links is 2710 kg/m³, while their cross-sectional area measures 6 x 10⁻⁵ m² Each link has a mass per unit length of 0.1626 kg/m and a moment of inertia of 4.5 x 10⁻¹¹ m⁴ Additionally, the Young’s modulus for the links is recorded at 7.11 x 10¹⁰ N/m².
The mass of the tip load in four cases mt (kg) 0; 0.05;
The driving torques are shown as Fig 3.5 and Fig 3.6
Figure 3 5 The torque at joint 1 Figure 3 6 The torque at joint 2
The values of joint variables of the link 1 and link 2 are described as Fig
3.7 and Fig 3.8 The values of the elastic displacements at the ending point of link 1 and link 2 are expressed as Fig 3.9, Fig 3.10, Fig 3.11 and Fig 3.12
The end-effector positions within the workspace for both rigid and flexible links are illustrated in Figures 3.13 and 3.14 Additionally, Table 3.2 presents the mass ratios between the flexible links and the tip load, highlighting the maximum elastic deformation.
Figure 3 7 The value of joint 1 variable
Figure 3 8 The value of joint 2 variable
Figure 3 9 The value of flexural displacement at the end of link 1
Figure 3 10 The value of slope displacement at the end of link 1
Figure 3 11 The value of flexural displacement at the end of link 2
Figure 3 12 The value of slope displacement at the end of link 2
Figure 3 13 The position of the end-effector in OX
Figure 3 14 The position of the end-effector in OY Table 3 2 The mass ratios between the flexible links and tip load
The mass of tip load (g)
The mass ratios: link/tip load
Table 3 3 The maximum elastic displacements at the ending points of the links
The links The mass of tip load (g)
Flexural displacement (m) 0.055 0.045 0.04 0.03 Slope displacement (rad) 0.14 0.11 0.09 0.07 Link
2 Flexural displacement (m) 0.021 0.013 0.014 0.018 Slope displacement (rad) 0.068 0.038 0.046 0.06
Some comments are can be drawn from the figures and the tabs above as follows
As the payload value rises, the elastic displacements at the endpoint of link 1 decrease while those at the endpoint of link 2 increase This results in larger vibration amplitudes at both endpoints, and the duration for the elastic displacements to return to zero is extended.
In scenarios where the mass ratio is approximately 4.81:1, such as a tip load of 50g, the time for elastic displacements to reach zero is minimized compared to cases without a payload This indicates a significant relationship between the mass of the links, the tip load, and the driving torques Lower elastic displacements at the end-effector are crucial for enhancing position accuracy in control systems Ideally, the mass ratio between the links and the tip load should be maintained within the range of 4:1 to 5:1 for optimal performance.
Analyzing the impact of payload variation on elastic displacements at the end-effector point highlights the critical importance of assessing the load capacity of robots, especially flexible ones The effectiveness of a payload can be quantified by the magnitude of elastic displacements and the duration required for these displacements to return to zero.
On the other hand, the variation of driving forces/torques and the length of flexible links is essentially studied when solving the optimal structure problem
2 The flexible robot type IV (P a R)
Previous studies have largely overlooked the impact of geometric parameters, particularly the length of flexible links, on elastic displacements at the end-effector point This section addresses these effects, providing valuable insights for selecting appropriate geometric parameters when designing flexible robots that incorporate various joint types.
Let us consider the flexible robot type IV as follows Fig 3.15
Figure 3 15 The flexible robot type IV
The parameters of system are shown in tab 3.4 The Fig 3.16 describes the schematic in SIMULINK solving the forward dynamic of the system
Figure 3 16 Schematic of solving forward dynamic in SIMULINK
The dynamic behaviors were examined by varying the link lengths in two distinct cases, as detailed in Table 3.5 The driving force and torque at the joints for both cases are illustrated in Figures 3.17 and 3.18.
Table 3 4 The parameters of the flexible robot type IV
The length of the rigid link 1 L1 (m) 0.2
The mass per length unit of the link 1 m1 (kg/m) 1.4 The inertial moments of the motor 1 and 2 Ih1=Ih2 (kg.m 2 ) 5.86x10 -4
The mass of the motor 2 mdc2 (kg) 0.16
The number of elements of the link 2 n2 3
The length of the link 2 L2 (m) 0.8
The length of each element of the link 2 l2e=L2/n2 (m) 0.8/3 The mass density of the link 1 and link 2 p1=p2 (kg/m 3 ) 7850
The width of the link 2 b (m) 0.04
The thickness of the link 2 h (m) 0.003
The cross-section area of the link 2 A=b.h (m 2 ) 1.2x10 -4 The mass per length unit of the link 2 m2 (kg/m) 1.05 The moment inertia of the link 2 I2 (m 4 ) 9x10 -11 The Young’s modulus of link 2 E2(N/m 2 ) 2x10 10
The mass of the tip load mt (kg) 0.1
Figure 3 17 The driving force rule Figure 3 18 The driving torque rule
Table 3 5 The length of the links in two cases
The length of the links Case 1 Case 2
L1 (m) 0.2 0.4 0.6 0.8 0.6 0.6 0.6 L2 (m) 0.6 0.6 0.6 0.6 0.6 0.8 1.0 Ratio L1/L2 1:3 2:3 1:1 4:3 1:1 3:4 3:5 a Simulation results of case 1
The values of the joint variables, illustrated in Figures 3.19 and 3.20, demonstrate a decrease as the length of link 1 increases, particularly when the mass of link 1 is elevated, while maintaining a constant driving force or torque.
Figure 3 19 The value of translational joint
Figure 3 20 The value of rotational joint
Figure 3 21 The value of flexural displacement
Figure 3 22 The value of slope displacement
Figures 3.21 and 3.22 illustrate the elastic displacements at the end-effector point, showing that maximum values occur simultaneously with the application of driving force and torque, rapidly decreasing to zero thereafter The differences in position deviations between the rigid and flexible models of the end-effector point along the workspace axes are depicted in Figures 3.23 and 3.24.
In general, the variations of the length of the rigid link 1 do not have much effect on the values of the end-effector point b Simulation results of case 2
As the length of flexible link 2 increases, its mass gradually rises Consequently, the variable value at joint 1 (see Fig 3.25) decreases less significantly compared to case 1, where the mass of link 1 is greater than that of link 2.
The value of the joint variable decreases as the mass and length of link 2 increase, as illustrated in Figure 3.26 The elastic displacements at the end-effector point, depicted in Figures 3.27 and 3.28, show a significant increase with the elongation of flexible link 2, resulting in a longer time required for these displacements to return to zero Additionally, the position deviations of the end-effector point along the workspace axes, shown in Figures 3.29 and 3.30, are notably large.
Figure 3 25 The value of translational joint Figure 3 26 The value of rotational joint
Figure 3 27 The value of flexural displacement
Figure 3 28 The value of slope displacement
Figure 3 29 The position deviation in OX
Figure 3 30 The position deviation in OY c Summary
The results of two case are summarized in tab 3.6 which shows the maximum values of the joint variables and the elastic displacements at the end- effector point
Table 3 6 The maximum values in two cases
Position control system of the planar serial multi-link flexible robots
Chapter 2 Dynamic modeling of the planar flexible robots
This chapter explores the kinematic and dynamic modeling of planar flexible robots featuring various joint types It establishes a general homogeneous transformation matrix and employs Finite Element Method (FEM) alongside Lagrange’s equations to formulate the dynamic equations Additionally, an extended assembly algorithm is introduced to construct the global mass and stiffness matrices, with its accuracy validated through comparisons with existing research.
Chapter 3 Dynamic analysis and position control of the planar flexible robots
This chapter addresses two key issues in the analysis of flexible robots: the forward and inverse dynamics, which are examined in relation to variations in payload, link length, and boundary conditions Additionally, an extended PID controller is developed to effectively manage the position of planar flexible robots The control law is established and rigorously validated using Lyapunov’s theory, while the controller parameters are optimized through a genetic algorithm.
EXPERIMENT
Objective and experimental model
This chapter details the execution of experiments aimed at verifying the calculation results for the forward and inverse dynamics of the flexible robot type IV, as outlined in chapters 2 and 3 The experimental data is derived from the actual robot type IV, and a comparison between the simulation results and the real model is essential to assess the accuracy of the dynamic modeling and the methods used to solve the motion equations.
Determining the values of the joint variables and the flexural displacement at the end-effector point is the objective to achieve the purpose of the verification
The flexible robot type IV is manufactured based on the parameters in tab 3.8 and is shown as Fig 4.1
DC motor; (2) Lead screw; (3) Encoder with step motor; (4) Step motor;
(5) Flex sensor; (6) Encoder with DC motor; (7) Flexible link
The design drawing can be found in Appendix 6 The translational joint Pa is powered by a DC motor (1) via a lead screw system (2), as illustrated in Fig 4.2, while the encoder (6) directly measures the rotary angle of the DC motor The rotational joint is operated by a step motor (4), depicted in Fig 4.3, with the encoder (3) providing direct measurements of the step motor's rotary angle Additionally, a flex sensor (5) is attached to flexible link 2 (7), connecting the first element to subsequent elements.
Figure 4 2 Lead screw system Figure 4 3 Step motor at the rotational joint The lead screw and motors are described as Fig 4.4, Fig 4.5 and Fig 4.6, respectively
Figure 4 4 Lead screw Figure 4 5 DC motor
* Parameters of the lead screw
Following the screw shaft, the dimension is 8(mm), the pitch is 8(mm), the length is 400(m) and the material is stainless steel
* Parameters of the DC motor GB 37-3530
The motor operates within a voltage range of 6 to 12 volts and features a gear ratio of 1:43.8, with a real rate of 251 RPM and a free run rate of 276 RPM It has a motor shaft dimension of 6 mm and delivers a maximum torque of 3 Nm.
* Parameters of the step motor NEMA 17
The current amplitude is 2(A) The step angle is 1.8 0 The shaft dimension is 5(mm) The maximum torque is 2.5(Nm) The holding torque is 0.82(Nm)
The pulse speed is 3200 (pulse/rev).
Parameters, equipment and method of measuring
The parameters measured include the pulse signals from the encoders and the deviation voltage of the flex sensor The translational joint variable is determined from the encoder measurements, which reflect the rotary angle of the DC motor and the lead screw Meanwhile, the rotational joint variable is calculated using the pulse signals provided by the encoder.
(3) The flexural displacement value is given by transforming the measurement results of the flex sensor
The measurement equipment includes the rotary encoders and the flex sensor
Figure 4 7 Encoder LPD3806 Figure 4 8 Flex sensor
The operating voltage is from 5(V) to 24(V) The revolution is 600 (pulses/rev) It has two phases A and B creating an electrical angle 90 0
Various sensors are employed to measure the deformation of elastic structures, including piezo sensors, strain gauges, capacitive sensors, and flex sensors Each sensor type offers distinct advantages and disadvantages, making them suitable for different applications in deformation measurement.
A piezo sensor, derived from the Greek word "Piezo," meaning pressure, is known for its accuracy but is often limited by higher costs and mechanical delays In contrast, a strain gauge measures resistance changes in response to applied forces, effectively converting force, pressure, stress, and strain Strain is defined as the deformation experienced by a material under force While strain gauges are highly accurate, they come with significant costs and necessitate advanced measuring equipment for optimal performance.
A capacitive sensor includes two conductive plates and depends on the environment between the plates a lot
The flex sensor, composed of a single layer of material, offers advantages such as low cost, water resistance, thermal stability, and ease of integration, making it a popular choice in various applications despite its lower accuracy and sensitivity compared to strain gauges Its ability to be reused multiple times adds to its appeal, particularly for small demand in the market This dissertation utilizes the flex sensor, which features a thin, special ink layer that alters its resistance based on the bending radius; resistance values range from 30 kΩ when unbent to 120 kΩ when bent at 90 degrees, with a risk of breakage if bent beyond this limit.
The limited value can reach 180 0 with the special flex sensor Denoted V in is the input voltage with range from 3(V) to 12(V) and V out is the output voltage
The parameters of flex sensor FSL0095-103-ST are shown as Fig 4.9 The limited working thermal is from -35 0 C to 80 0 C The limited resistance is from
R u (not deformed) to R lim = 110(k ) (maximum deformed) The input voltage is V in = 4.98( ) V
Figure 4 9 Flex sensor FSL0095-103-ST 4.2.3 Measurement method
The motors, encoders, and flex sensor are linked to the Arduino 2560 terminal board, which facilitates programming of the power supplies, signal processing, and result display Detailed program codes can be found in Appendix 5.
The Fig 4.10 describes the diagram of the system which connects the equipment executing the experimental requests
Figure 4 11 Principle diagram inside Arduino 2560
The terminal board Arduino 2560 warrants to transmit the control signals to the
The control of the DC motor is managed through the L298 circuit, while the TB6560 circuit controls the step motor The terminal board facilitates signal reception from the flex sensor and encoders, handling all signal transmission and reception directly LABVIEW software interfaces with Arduino to display results, which are then exported to an EXCEL format file Figure 4.11 illustrates the principle diagram utilized within the Arduino system.
2560 connecting with other equipment Fig 4.12 shows the diagram in LABVIEW software.
Figure 4 12 LABVIEW diagram 4.4 Experimental orders
The experiments are executed following steps below
Step 1 Test and evaluate the stability of the system
This process utilizes pre-prepared calculators to assess system stability by measuring various parameters through multiple tests, aimed at identifying and eliminating manufacturing and assembly errors Official experiments are conducted only after confirming the system's stability.
- Run the application software Arduino 1.8.5 to translate the program codes and transmit the data to the processor Arduino 2560
- The LABVIEW is used simultaneously to read input and write output signals The Hercules 3 control the actuators
Step 3 Handle the signals to display on LABVIEW software
4.5 Method of handling the measurement data 4.5.1 Calculating the input signals
The input signals include the voltage into DC motor, the pulse signals into step motor
1 Calculating the input signal into DC motor GB37 3530
Assumed that the translational joint is moved a distance of d mm 1 ( ) in time
The lead screw, with a pitch of p mm, is connected to a DC motor via a coupling, ensuring that the rotation of the motor shaft directly translates to the rotation of the screw shaft Consequently, for every rotation of the motor shaft, the translational joint moves a distance of p mm, requiring the motor shaft to rotate by an angle of 2π radians.
in t s f ( ) warranting distance d mm 1 ( ) The angular velocity of DC motor is 2 1
Besides, the DC motor GB37 works at two voltage levels which are low level (6V) and high level (12V) with the angular velocity is predetermined
2 Calculating the input signals into step motor NEMA-17
During the period from 0 to tf, the rotational joint rotates by an angle of q2 radians, coinciding with the movement of the translational joint Since flexible link 2 is directly attached to the step motor shaft, the angle q2 also represents the rotation of this shaft within the same timeframe The step angle for the motor is 1.8 degrees.
3200 (pulse/rev) So, the number of pulse signals are 2000q 2
to warrant the angle q rad( ) in t s( ) The value of time to provide a pulse signal into the step motor is
In experiments addressing the inverse dynamic problem, it is important to recognize that the predetermined joint variables are time-dependent functions Consequently, the input signals to the motors are aligned with the frequency or cycle of these joint variables.
1 Handling the signals of encoder of DC motor
The number of pulses received from encoder which is fixed to DC motor at time step t i is a i Each a revolution (360 0 ) of the DC motor shaft corresponds
600 pulses So, the value of angle is (0.6 )a i 0 respectively a i pulses The value of translational joint variable is 1 ( ) 0.6 (mm)
2 Handling the signals of encoder of step motor
The output signals from stepper motor encoders are processed similarly to those from DC motor encoders At each time step \( t_i \), the number of pulses received is denoted as \( b_i \) The corresponding value of the rotational joint variable is 2.0, specifically 0.6.
3 Handling the measurement of the flex sensor
The output signal of flex sensor is the analog signal S a [117], [118], [132]
The analog signal converter of the processor Adruino 2560 is 10(bit) respectively the value (2 10 − =1) 1023 The output voltage can be converted as
Following [117], [132], the output voltage is rewritten as
V , (4.3) where, R 1 is the flat resistance and R 2 is the bend resistance value The flex sensor circuit is shown as Fig 4.13
The value of bend angle flex of flex sensor is determined as [132]
Substituting the eq (4.2) and eq (4.3) into the eq (4.4), we have:
= − −− (4.5) Assuming the small deformation, the value of the flexural displacement at the end-effector point can be approximately determined as
4.6 Experimental results 4.6.1 Forward dynamic experiment
Figures 4.14 and 4.15 illustrate the driving force and torque rules at the joints for calculation purposes The voltage and pulse signals sent to the motors are generated by translating these driving rules through the program codes detailed in Appendix 5.
Figure 4 14 Driving force Figure 4 15 Driving torque
The values of joint variables between the calculations and experiments are given in Fig 4.16 and Fig 4.17
Figure 4 16 The value of translational joint variable
The maximum deviation observed in the translational joint variables is 4.35%, while the rotational joint variables show a deviation of 7.5% across two cases These discrepancies can be attributed to approximations made during calculations and real-world conditions.
Figure 4 17 The value of rotational joint variable
Figure 4 18 The value of flexural displacement
Figure 4.18 illustrates the flexural displacement values at the end-effector point across two scenarios, revealing a maximum deviation of approximately 7.5% These findings confirm the accuracy and applicability of the dynamic equations and the method used for solving derivative equations in practical situations.
The laws of joint variables are given as
The results of the calculations were obtained using the inverse dynamics method outlined in Chapter 3 Experimental outcomes were derived from the laws of joint variables, which were translated into voltage and pulse signals for the motors This conversion process was carried out on an Arduino 2560 processor, with the program codes detailed in Appendix 5.
Figure 4 19 The value of translational joint variable