Recent Advances in Mechanical Engineering and Mechanics
A Multi-Joint Single-Actuator Robot: Dynamic
and Kinematic Analysis
A.Nouri, and M.Danesh.
joint axis has a non zero angle with respect to the others,
however, all joints axes of the outer group are in the same
plane, i.e., each joint axis has a zero angle with respect to the
others. If the inner group is rotated, the difference in its joints
angles makes the mechanism to bend merely in the desired
joint. For example, for bending the third joint, the inner group
must rotate until the angle between the third inner joint axis
and the third outer one becomes zero. On the other hand, it
can’t bend in the other joints (e.g. in the fourth joint) because
there is a non zero angle between the other inner joint axes
and their corresponding outer ones.
Abstract- Changing the length of the arm leads us to a new
approach; we name it multi-joint single actuator. To find out the
control approach, the mechanism is described in this article. Using
one actuator for lots of joints and each time connecting one of them
to this actuator is required 2 issues. The first issue is to control the
freedom of joint. Another is transferring the torque to the desired
joint. In this article, a mechanism is proposed for satisfying both. For
the second issue, two approaches are proposed and analyzed.
Equations for both of are found.
Keywords—Robot, mechanism, joint, control, arm.
I.
For manufacturing this robot, one polyamide cylinder was
located inside the other one. Then both of them were cut in the
same length. The links were built and joined together, as
shown in Fig.1. Then the inner joints were put inside the outer
ones.
INTRODUCTION
M
ECHANICAL arm [1] is a robot act like human arm
and, is used in different functions like welding, painting,
assembling and rehabilitation [2]. In these robots each joint
has its own actuator and controller. For controlling several
degrees of freedom, same number of actuators is needed like
snake robots [3, 4, 5] which have several actuators. Another
kind of robots is continuous robot [6, 7] which consists of
three actuators. They move a continuous arm like elephant
nose. Joints are not controllable separately in continuous
robots. Controlling several joints with one actuator is an issue,
is investigated in this paper. Two mechanisms is proposed for
this purpose. The first one is for selecting the desire joint and
other one for transferring the torque to that joint. For
transferring the force to the joints there is two ways. The first
one is wire and some springs and the second one is using a
series of gears. In this article mechanical design and dynamic
analysis of both is proposed but gears is used for transferring.
This robot can be used in many applications which there are
several joints and each time one of them is moving. This
mechanism helps us to reduce the number of actuators to 3 for
several joints. The only factor that limits us in number of
joints is the accuracy of our devises like our manufacturing
machineries and our actuator. One of the application of this
mechanism is in endoscopy [8] and Laparoscopic [9] since the
robot can work in different workspaces. By improving this
mechanism in future works we can use it in multi actuator
robots like humanoid robots [10].
II.
To rotate the desired joint by exerting the actuator force, we
investigate two approaches. The first approach is based on
using the wire and spring. A wire is along the robot and one of
its ends is connected to the end of the robot and its other end is
connected to the actuator. When the actuator pulls the wire, its
length decreases and makes the robot to rotate. On the other
side of the robot (the side with no wire), there are some
springs. If the pulling force of the wire is removed then these
springs release and the robot rotates to the other side. This is
similar to the method used in some continuous robots.
MECHANICAL DESIGN
The proposed mechanism consists of two coaxial groups of
joints, inner group and outer group. In the inner group, each
ISBN: 978-1-61804-226-2
Fig.1 angle of inner and outer axis
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Recent Advances in Mechanical Engineering and Mechanics
The second approach is based on using a series of gears which
the last gear attached to the robot body and the first gear is
connected to the actuator shaft. All of the other gears are free
to turn and some of them are located on the joints (one gear on
Fig.3. A 3D coordination
(1)
(2)
A 3D coordination is shown in Fig.3. x axes are the same in
both recent figures, however, y axes are different in them. The
end effecter position in this coordination is
Fig.2 Cartesian coordination
each joint).For example, if the third joint is allowed to move,
the gear on this joint acts like a gear attached to the third link.
When the actuator’s shaft rotates, only the gears before the
third joint rotate. This leads to the arm movement in the third
joint.
(3)
(4)
(5)
Returning the spring to its normal extension is not
controllable, thus, the second approach is more accurate than
the first approach. On the other hand, in the second approach
the ratio of gears can be chosen by designer to control the
velocity and the force of each joint.
III.
The relationship between and other variables can be derived
geometrically. Fig.4 shows the geometrical model of this
robot.
a is the perpendicular distance between the wire and the link
axis, and b is the half of the distance between two adjacent
joints.c is the hypotenuse of the right-angled triangleabc. Fs
shows the spring force of the joint i. Fc is the wire force
supplied by the actuator. is the angle between the two
adjacent c hypotenuses in the wire side. β is the same angle,
however in the spring side. and β are calculated by the
following equations.
DYNAMIC ANALYSIS
A. ROBOT WITH WIRE AND SPRING
KINEMATIC
Consider Cartesian coordination defined in the plane that the
robot turns in it, according to Fig.2.
In Fig.2, link i is the last movable link and n is the total
number of robot links. is the angle between the movable link
axis and horizontal axis, x. (x,y) shows the position of the end
effecter. is limited by the structure. Due to the links structure
in this prototype, the limit is 60 degree. The end effecter
position is derived as
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Recent Advances in Mechanical Engineering and Mechanics
Fig.5. Typical characteristics of different kinds of springs
Spring force in each group of springs is calculated by the
following equations.
Fig.4. different angles in spring and wire
(7)
hard
Liner
soft
(10)
n>1
In this robot, xs is calculated by
The spring and wire torques are
(8)
To compute the moment of inertia in each mode of robot, a
Catia model for each link is developed. Based on this model,
the moment of inertia of each link around its z axis (Jn) and
then the moment of inertia of all movable links around the
movable joint are calculated as
The joint torque is
(9)
(11)
There are three different kinds of springs, hard, soft and linear,
and their typical characteristics are shown in Fig.5.
M is the mass of movable joints and m is
the mass of each link.
(12)
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Recent Advances in Mechanical Engineering and Mechanics
From the following equations
(16)
(17)
Then,
(18)
(19)
(20)
(21)
Fig.6. forces and moments in robot
DYNAMICS
Kinematic energy comes from the rotation of arm.
Dynamic analysis can be done with both Newton-Euler
and Lagrangian.
(22)
NEWTON-EULER:
Then
In the following formulation, the torque is calculated. We
rewrite all of the formulation with ө.
(23)
(13)
At last Q is the generalized force.
LAGRANGIAN:
(24)
In this part, the robot dynamic relations are derived by
Lagrangian.
From Lagrangian formula and its ingredients, we have
(14)
(25)
In this robot, we have 2 kinds of potential energy, spring
and gravity.
(15)
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Recent Advances in Mechanical Engineering and Mechanics
description
Length
Force
Force
Mass of each link
Gravity
symbol
L
FX
FY
m
g
value
0.05
2
1
0.117
9.81
Angle
π/2
Links number
Moment of inertia
i
j
1
0.00004149
Table 1. Robot characteristics
For gears, it is easier to find the desired arm torque.
Because the actuator moment is directly transferred to the
joint and can be used directly in the formulation, thus, we
rewrite the equations as
B. ROBOT WITH GEARS
KINEMATICS:
Robot kinematic with gear is the same as that of with
spring and wire.
Newton Euler:
(26)
Here, the potential energy includes only gravity term.
(29)
(27)
(30)
Lagrangian:
(28)
(31)
Therefore, the dynamic equation is as
(26)
DYNAMIC ANALYSIS:
IV.
THEORITICAL RESULTS
We simulate all the equations in MATLAB and get these plots
as the result
(a)Torque(n.m) plot for joint 1
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Recent Advances in Mechanical Engineering and Mechanics
(b)Torque(n.m) plot for joint 2
(d)Torque (n.m) plot for joint 4
Fig.6. Theoretical results, (0)=-pi/3,
= (0) +t pi/10;
V.
EXPERIMENTAL RESULTS
To find out the effectiveness of the proposed robot design, as
it is depicted in fig.7, we made this robot and tested the
mechanism. The robot parameters are mentioned in table 1.
All of the robots links are made of polyamide which is suitable
for production procedure of this
(c)Torque(n.m) plot for joint 3
Fig.7. the prototype of robot
prototype. Polyamides are very hard and need to be cooled in
milling process.
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In this robot, a RX-64 motor is used for turning the arm in
each case. This motor is controlled via MATLAB (software)
and is commanded through USB2DYNAMIXEL connector
(hardware interface). All of the feedback data are stored in a
matrix and are plotted.
Fig.8. presents a comparison between the actual torque and
theoretical torque for the same trajectory. Although we tried to
consider all factors in the theoretical calculation, however, we
can’t ignore the friction and some production faults effects. In
this mechanism, manufacturing accuracy is very important. If
inner and outer joints are not exactly coaxial then some
difficulties appears in turning the inner joints. Also the axes of
the inner joints and outer joints must be in the same plane. In
the case, the angle between the axis of the 3rd inner joint and
the axis of the 3rd outer joint becomes zero, if they do not
coincide, they won’t move. If the joints have backlash or free
space, it causes unwanted turn in other joints.
(c)Torque (n.m) plot for joint 3
(d)Torque (n.m) plot for joint 4
Fig.8. experimental results, (0) = -pi/3,
= (0) +t pi/10;
(a)Torque (n.m) plot for joint 1
V.
CONCLUSION
3B
In this paper, a novel mechanism is presented to robot design,
manufactured and practically tested. This robot has the
following contributions.
1) Several joints can be controlled with just one actuator
and turned with only one actuator; however, in a
simple arm an actuator is needed to move each joint.
2) The work space of a continuous arm or a simple
robot arm is limited. In this robot, there is a work
space for each mode.
3) The mechanism, used in this robot can be used in
other robots, especially if there is a need for
adjustable joints.
4) Changing the radiuses of gears in this robot can
transfer different torques to the joints and it is useful
(b)Torque (n.m) plot for joint 2
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when we want different torques in each link with the
same actuator.
In this robot, the accuracy in the production process is very
important. In comparison with a robot with the same number
of joints, this robot costs less.
Acknowledgment
This study is partially supported by Isfahan University of
Technology.
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