key: cord-0303067-x09h7o31
authors: Woodside, Mitchell R.; Bristow, Douglas A.; Landers, Robert G.
title: A Kinematic Error Observer for Robot End Effector Estimation
date: 2020-12-31
journal: Procedia Manufacturing
DOI: 10.1016/j.promfg.2020.05.145
sha: 2bbab504f6c8eeada1acea827e594e003e861d7c
doc_id: 303067
cord_uid: x09h7o31

Abstract Industrial robots were initially designed to be low cost and highly repeatable for pick-and-place and assembly operations. As a result, their construction does not support the high-precision accuracy and bandwidth needed for more advanced industrial automation. Due to the growing interest in replacing high precision manufacturing equipment with industrial robots for some applications, external high-precision feedback sensors (e.g., laser trackers), are needed to improve robot position and orientation accuracy. Before external sensors can be combined with control methodologies to improve robot accuracy, it is important to properly condition the measurement signals. In this paper, a Kinematic Error Observer (KEO) is derived and used to estimate a robot’s slowly changing kinematic errors in real-time. An analysis of the KEO is first conducted to demonstrate its stability properties, transient response characteristics, noise sensitivity, and performance in non-deterministic measurement systems. Next, an implementation of the KEO was used in conjunction with a laser tracker and 6DoF sensor to dynamically estimate the kinematic errors of a Yaskawa/Motoman MH180 industrial robot. The results show that the KEO is capable of estimating the robot’s quasi-static kinematic errors in both static and dynamic environments. It was also found that the KEO, with small enough observer gain, attenuates high frequency measurement noise and produces a smooth estimation of the robot’s kinematic error.

There has been a recent increase in nationally funded research initiatives [1, 2] , industry-academic research collaborations [3] , and the rate of annual academic publications [4] on robotic machining. It has been reported by several researchers [4] - [7] that the sale of industrial robots in the last few years has been steadily increasing; however, only 1.5 -5% of industrial robots are being used for machining applications (milling, drilling, … etc.).

The increase in industrial and academic research, coupled with the small percentage of industrial robots being utilized for robotic machining applications, suggests that there is a growing need to improve the implementation and theory for the use of robots in high-precision industrial applications, including machining. The combined efforts of many researchers [8] - [13] have helped to determine the primary drawbacks which are preventing this adoption:

• low frequency vibrations of the industrial robot due to both low and varying stiffness properties of the robot structure, • poor absolute accuracy due to discrepancies between the physical robot and its kinematic model, and • low bandwidth interfacing of external sensors to industrial robot controllers. Previous attempts to find a solution to these drawbacks have included the use of stiffness optimization [14] - [17] , chatter avoidance [18] - [20] , active vibration damping [21] - [23] , force feedback control [24, 25] , positional feedback control [26, 27] , and improving controller hardware for higher bandwidth sensor input [25, 27, and 28] . Only a few publications [16, 26, and 29] have used in-loop metrology, such as laser trackers, as the external sensors in their feedback loops. The high accuracy and limited obtrusiveness of laser trackers makes them a viable solution for external sensor feedback control in robotic manufacturing applications.

As a first step toward feedback control, this paper considers the problem of dynamically estimating robot kinematic errors,

There has been a recent increase in nationally funded research initiatives [1, 2] , industry-academic research collaborations [3] , and the rate of annual academic publications [4] on robotic machining. It has been reported by several researchers [4] - [7] that the sale of industrial robots in the last few years has been steadily increasing; however, only 1.5 -5% of industrial robots are being used for machining applications (milling, drilling, … etc.).

The increase in industrial and academic research, coupled with the small percentage of industrial robots being utilized for robotic machining applications, suggests that there is a growing need to improve the implementation and theory for the use of robots in high-precision industrial applications, including machining. The combined efforts of many researchers [8] - [13] have helped to determine the primary drawbacks which are preventing this adoption:

• low frequency vibrations of the industrial robot due to both low and varying stiffness properties of the robot structure, • poor absolute accuracy due to discrepancies between the physical robot and its kinematic model, and • low bandwidth interfacing of external sensors to industrial robot controllers. Previous attempts to find a solution to these drawbacks have included the use of stiffness optimization [14] - [17] , chatter avoidance [18] - [20] , active vibration damping [21] - [23] , force feedback control [24, 25] , positional feedback control [26, 27] , and improving controller hardware for higher bandwidth sensor input [25, 27, and 28] . Only a few publications [16, 26, and 29] have used in-loop metrology, such as laser trackers, as the external sensors in their feedback loops. The high accuracy and limited obtrusiveness of laser trackers makes them a viable solution for external sensor feedback control in robotic manufacturing applications.

As a first step toward feedback control, this paper considers the problem of dynamically estimating robot kinematic errors,

There has been a recent increase in nationally funded research initiatives [1, 2] , industry-academic research collaborations [3] , and the rate of annual academic publications [4] on robotic machining. It has been reported by several researchers [4] - [7] that the sale of industrial robots in the last few years has been steadily increasing; however, only 1.5 -5% of industrial robots are being used for machining applications (milling, drilling, … etc.).

The increase in industrial and academic research, coupled with the small percentage of industrial robots being utilized for robotic machining applications, suggests that there is a growing need to improve the implementation and theory for the use of robots in high-precision industrial applications, including machining. The combined efforts of many researchers [8] - [13] have helped to determine the primary drawbacks which are preventing this adoption:

• low frequency vibrations of the industrial robot due to both low and varying stiffness properties of the robot structure, • poor absolute accuracy due to discrepancies between the physical robot and its kinematic model, and • low bandwidth interfacing of external sensors to industrial robot controllers. Previous attempts to find a solution to these drawbacks have included the use of stiffness optimization [14] - [17] , chatter avoidance [18] - [20] , active vibration damping [21] - [23] , force feedback control [24, 25] , positional feedback control [26, 27] , and improving controller hardware for higher bandwidth sensor input [25, 27, and 28] . Only a few publications [16, 26, and 29] have used in-loop metrology, such as laser trackers, as the external sensors in their feedback loops. The high accuracy and limited obtrusiveness of laser trackers makes them a viable solution for external sensor feedback control in robotic manufacturing applications.

As a first step toward feedback control, this paper considers the problem of dynamically estimating robot kinematic errors,

There has been a recent increase in nationally funded research initiatives [1, 2] , industry-academic research collaborations [3] , and the rate of annual academic publications [4] on robotic machining. It has been reported by several researchers [4] - [7] that the sale of industrial robots in the last few years has been steadily increasing; however, only 1.5 -5% of industrial robots are being used for machining applications (milling, drilling, … etc.).

The increase in industrial and academic research, coupled with the small percentage of industrial robots being utilized for robotic machining applications, suggests that there is a growing need to improve the implementation and theory for the use of robots in high-precision industrial applications, including machining. The combined efforts of many researchers [8] - [13] have helped to determine the primary drawbacks which are preventing this adoption:

• low frequency vibrations of the industrial robot due to both low and varying stiffness properties of the robot structure, • poor absolute accuracy due to discrepancies between the physical robot and its kinematic model, and • low bandwidth interfacing of external sensors to industrial robot controllers. Previous attempts to find a solution to these drawbacks have included the use of stiffness optimization [14] - [17] , chatter avoidance [18] - [20] , active vibration damping [21] - [23] , force feedback control [24, 25] , positional feedback control [26, 27] , and improving controller hardware for higher bandwidth sensor input [25, 27, and 28] . Only a few publications [16, 26, and 29] have used in-loop metrology, such as laser trackers, as the external sensors in their feedback loops. The high accuracy and limited obtrusiveness of laser trackers makes them a viable solution for external sensor feedback control in robotic manufacturing applications.

As a first step toward feedback control, this paper considers the problem of dynamically estimating robot kinematic errors, 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to i.e., the difference between the calculated, based on joint measurements, and actual location and orientation of the end effector. Kinematic errors arise from errors in the robot models used to calculate end effector position and orientation, such as link lengths, joint offsets, backlash, etc. Further, quasi-static kinematic errors arise from end effector load. Measurement of kinematic errors at static positions can be obtained in a straightforward manner using an external measurement system; however, for real-time estimation the problem is challenging due to limitations in synchronization between the robot feedback and the measurement system.

In Section 2 the kinematic error of an industrial robot is defined and calculated. Then, a Kinematic Error Observer (KEO) for real-time position and orientation estimation is derived for a quasi-static system. In Section 3 KEO stability, performance and sensitivity analyses are conducted, as well as a timing analysis which highlights the effects of the nondeterministic measurements used with the KEO. In Section 4, step response and constant velocity motion experiments are conducted to demonstrate its performance. Finally, in Section 5, the paper is summarized, and conclusions are drawn. 

In this section a robot kinematic error model is presented, details regarding the measurement of the kinematic errors are given, and the derivation of the Kinematic Error Observer (KEO) is presented. 

In order to identify the physical errors within an industrial robot's kinematic structure, many publications [30] - [33] model kinematic error as an integral component of the robot's forward kinematic chain. Hence, components of the kinematic error are typically defined relative to each link in the chain, and the total kinematic error is usually evaluated in reference to the robot's base frame. Since the goal of the KEO is to create a real-time estimation of the kinematic error through external sensor feedback, it is not necessary for kinematic errors to be modeled as structural components of the robot's forward kinematic chain.

Another approach is to measure the physical location of the end effector (referred to in this paper as the sensor frame) with respect to the end effector nominal location (referred to in this paper as the tool frame). The relative transformation between these two frames describes the transformative error between the physical robot and the robot's kinematic structure. This dynamic error transformation will change both throughout the robot's motion (i.e., it is position-dependent) and from external disturbances acting on the robot. Figure 1 shows a graphic depiction of this transformation.

The dynamic error transformation from the tool frame to the sensor frame is

where the dynamic error transformation's rotation matrix, is the position of the sensor frame with respect to the tool frame.

For implementation, the measured position and orientation kinematic errors are combined into a single vector. To accomplish this, the axis-angle representation,

of the error transformation's rotation matrix is used, where t s r is the axis-angle representation and the function 3 3 3 : x a f → computes the axis-angle representation from the rotation matrix. The calculations which define this function are described in the Appendix. Together, the vectors t s p and t s r , which represent the position and orientation of the sensor frame with respect to the tool frame, define the industrial robot's measured kinematic error

The sensor frame is measured by an external sensor (such as a laser tracker), and the nominal tool frame is calculated from the industrial robot's forward kinematics, which are a function of the robot's measured joint positions, as calculated by the encoder readings. The error transformation is then calculated as the relative transformation between these two frames and the measured kinematic error vector can be derived from the error transformation.

Consider, an n-link industrial robot's nominal forward kinematic chain,

where the vector of measured joint positions is,

and the robot's nominal forward kinematics defines the nominal transformation of the robot's n th frame with respect to its base frame for the vector of measured joint positions. The modeling technique is described in more detail in [34] . The transformation to the sensor frame with respect to the robot's base frame is then

where n t T is the static transformation of the tool frame with respect to the robot's n th frame. The rotation matrix are the orientation and position of the sensor frame with respect to the robot's base frame, respectively.

The transformation from the robot's base frame to the tool frame is

where the rotation matrix

are the orientation and position of the tool frame with respect to the industrial robot's base frame, respectively. Both are a function of the measured joint positions. Using (7) and (8), the transformation from the robot's base frame to the sensor frame becomes

Solving (9) for the error transformation is

Combining (1), (7), (8) , and (10) ( ) ( ) ( )

From (3) and (11) the position and orientation sub-vectors of the measured kinematic error, defined in (4), are ( ) ( )

respectively.

Consider the measured system output,

where x is the system's state vector and n is sensor noise. The observer dynamics are given by

where L is the observer gain. Note that since the system is modeled as quasi-static, the nonlinear kinematic mapping does not appear in equation (15) . In this work, the system output is,

where 6 m  e is the measured kinematic error vector given in (4), and the estimated output vector is

where the vector 6  e is the estimated kinematic error. Combining (15) , (16) , and (17), the KEO is

We will assume that the noise, n, is uncoupled across each channel and, thus, select an observer gain of the form, ( , , , ) diag L L L = L .

The transfer function relating the estimated kinematic error to the measured kinematic error is

The transfer function implies that L > 0 will produce a stable estimation of the measured kinematic error. The time constant of the first order KEO is

As the observer gain is increased, the settling time decreases.

It is likely that a form of noise will be present in the measured kinematic error signal. This miscellaneous noise may come from the measurement sensor, excited robot dynamics, and non-deterministic timing. In the presence of sensor noise, the quasi-static state in (14) is,

where 6  e is the true kinematic error. The relationship between the KEO's estimated kinematic error, true kinematic error, and measurement noise can be found by combining (14) , (18) , and (21),

To study the noise sensitivity of the KEO as a function of the observer gain, a simulation of the KEO was constructed from (22) with miscellaneous noise simulated as a normal distribution,

where n  is the standard deviation of the miscellaneous noise.

For all simulations, n  and L were iterated over the ranges of 0-0.5 (mm, rad) and 1-20, respectively. The simulation was conducted at the joint position in Table 1 . Post simulation, the change in standard deviation of the estimated kinematic error (output) and the miscellaneous noise (input) were used to calculate the noise sensitivity of the KEO for different values of observer gain. Due to the linear relationship (22), the noise sensitivity of the KEO was observed to be constant for each value of observer gain. The resultant constant noise sensitivities for the simulated values of gain are shown in Table 2 . In Table 2 , it is shown that as the gain increases, the KEO's sensitivity to noise also increases. This is due to the improved transient response characteristics of the KEO (i.e., the time constant decreases as gain increases). By increasing the gain and improving the transient response, the KEO is able to estimate higher frequencies of the input which results in higher frequencies of measurement noise being propagated through the system. Therefore, the value of observer gain should be chosen to reflect either high frequency noise rejection or faster transient response characteristics.

The results of the stability and transient response analyses demonstrate that a positive observer gain will produce a stable estimation and that as the gain increases the settling time decreases. The results of the sensitivity analysis show that as the observer gain increases, more measurement noise will propagate through the system. These analyses provide guidelines as to the tradeoffs when increasing and decreasing the observer gain.

For the studies conducted in this paper, the sensor frame and robot encoder measurements were not synchronized. The nondeterministic timing was modeled by a varying time delay,

where o t is the average timing delay between the sensor frame and encoder measurements and t  is the non-deterministic component of the timing delay, modeled as a normal distribution,

where t  is the standard deviation of t  .

The variable timing delay will only affect the kinematic error calculation in (12) and (13) if the robot is in motion. This is due to the variable timing delay causing the measurement of the sensor frame and the robot encoders to be taken at different positions. To quantify this timing dependent error, let the variable timing delay be applied to the measurement of the sensor frame. The time delayed position (12) and orientation (13) , respectively, of the measured kinematic error vector become

The time delayed position , respectively, of the sensor frame measurement with respect to the robot base frame while undergoing constant position and angular velocity can be approximated as

where the terms

are the positional and orientational error, with respect to the robot's base frame. Using the approximations in (28) and (29) with (26) and (27), the approximate timing dependent errors, respectively, are

Using (36) the KEO used in a non-deterministic measurement system is

If the average timing delay does not drift significantly during operation, it is possible to reduce ( ) t t v by aligning the sensor and encoder measurements (i.e. 0 ( ) 0

). It should be noted that the non-deterministic timing noise, ( ) t n , will always be present in the measurement.

The timing delay can be found experimentally by commanding the robot in a single spatial direction at a constant velocity and calculating the timing delay by

where i e is the constant offset of the measured kinematic error that is observed in the spatial direction i and b i v is the commanded velocity with respect to the base frame in the spatial direction i . If the measurements are aligned, (37) reduces to,

which is the time dependent representation of (22) . This implies that the KEO used in a non-deterministic measurement system (i.e., with zero average time delay between signals) will perform similarly to the KEO derived in Sections 3.1 and 3.2. Therefore, a costly deterministic measurement system may not be necessary to accurately estimate the robot's kinematic errors.

The system used to evaluate the performance of the KEO was comprised of a Yaskawa/Motoman MH180 industrial robot with a DX200 robot controller, an Automated Precision Inc. (API) Radian laser tracker, an API SMARTTRACK (STS) 6DoF sensor, a Windows 10 PC (used to collect the API measurement data), and an Ubuntu Linux computer running the Robot Operating System (ROS). Table 3 gives the specifications for all equipment used in the experimental setup and Fig 3 shows the experimental setup. 6 Author name / Procedia Manufacturing 00 (2019) 000-000 Figure 2 shows a block diagram of the experimental system. Joint commands are sent to and executed by the DX200 robot controller via the teach pendant. The MotoROS binary application loaded on the DX200 controller sends timestamped encoder measurements at measured frequency of 39.56.0 Hz to ROS running on the Linux computer (via TCP/IP). The laser tracker and 6DoF STS collect sensor frame measurements using an API SDK and sends them through the windows computer via TCP/IP to the Linux computer at a measured frequency of 99.615.5 Hz. The ROS stores each measurement in an individual buffer and extracts the most recent measurement at a programmed frequency of 20 Hz, which is the greatest common multiple between measurement systems, in order to roughly correlate the two measurements. Once the encoder and sensor frame measurements are extracted from the buffer, the kinematic error is calculated, and then estimated by the KEO. The use of TCP/IP, two computers, un-synchronized sampling rates, and the lack of a real-time OS makes the experimental system non-deterministic.

In this experiment the transient and steady-state performance of the KEO to a step response was evaluated by commanding the robot to a single location (Table 4 ) and initializing the KEO with zero-initial condition. This experiment was repeated for observer gain of L = 1, 5, and 10. Table 5 shows the experimental 2% settling times and corresponding theoretical values. As shown in Table 5 , the percent difference between the theoretical and experimental 2% settling times 8.44-16.7%, which is similar to the approximately 15% variations in the Author name / Procedia Manufacturing 00 (2019) 000-000 measurement systems. The difference in the theoretical and experimental settling times dramatically increases for L = 10. This is likely due to the fact that the time constant for this observer gain is 0.1 s, which is only four times the sample period (i.e., 0.025 s). To avoid large deviations from the model, an observer gain of L = 1 was used in the experimental results. 

In this experiment, the response of the KEO was evaluated as the robot was commanded vertically between three positions at a constant velocity of 100 mm/s (see Figure 4 ). At position 1, the KEO was initialized with zero estimated kinematic error and a 4 s delay was used to allow the KEO to stabilize on the measured kinematic error. After the delay, the robot was commanded to position 2. Another 4 s delay was executed to allow the KEO to stabilize. After the second delay, the robot was commanded to position 3, terminating the experiment. In Figure 5 , the solid red, blue, and green lines show the x, y, and z position of the nominal tool frame with respect to the robot base frame, which is shown in Figure 4 , calculated from the encoder measurements. The dashed red, blue, and green lines show the x, y, and z position of the sensor frame with respect to the base frame, measured with the laser tracker. As observed in the figure, there is a timing delay between the encoder measurement and the laser tracker measurements during the constant velocity experiment. Figure 6 shows the resultant measured and estimated kinematic error during the constant velocity experiment. In this figure (and subsequent figures) the solid red, blue, and green lines are the x, y, and z components, respectively, of the measured kinematic errors (both position and orientation). The dashed red, green, and blue lines are the x, y, and z components, respectively, of the estimated kinematic errors. The 1, 2, and 3 in the upper figure denote the corresponding robot position from Figure 4 where the motion was paused while the KEO stabilized.

In Figure 6 , while the robot is in motion, both a constant offset and timing noise are visible in the z component of the measured positional kinematic error. These errors are due to the non-deterministic time delay between the laser tracker and encoder measurements show in Figure 5 and discussed in Section 3.3. Since the robot's motion was vertical and aligned to the tool frame's z-axis, timing errors are only present in the z component, as predicted by the timing analysis (34) .

The polarities of the constant offsets are determined by if the laser tracker measurement is leading or lagging the encoder measurement. If the laser tracker measurement is lagging, the constant kinematic error offset is negative. If the laser tracker measurement is leading, the constant kinematic error offset is positive. This is confirmed by (34), Figure 5 , and Figure 6 .

Another interesting observation in Figure 6 is the x component of the measured positional kinematic error. Notice that the measured x component changes slowly throughout the robot's motion. Based on previous work [31] [32] this is the expected dynamics of the kinematic error, and it is observed that the KEO is capable of estimating the kinematic error in a dynamic environment. Author name / Procedia Manufacturing 00 (2019) 000-000 Due to the fact that the robot's orientation was relatively constant, the timing errors observed in Figure 6 are not present in the orientation components of the kinematic error (40). However, there are several small oscillations in the measured kinematic error. These correlate to the instances when the robot executed its motion to the next position. These oscillations are likely due to small excitations in the robot's dynamics when the torque suddenly changes. Examining the magnification in Figure 7 , regardless of their origin, the KEO filters out the oscillations and provides a smooth estimation of the kinematic error.

The largest measurement error in Figure 7 is due to the z component of the orientational kinematic error. Since this error is present at the beginning of the experiment (i.e., when the robot is not moving) it is believed to be due to inaccurate modeling of the nominal tool frame with respect to the robot's n th frame. It should be noted that once the KEO settles on this error, it tracks the error throughout the duration of the experiment.

As discussed in Section 3.3, the average timing delay can be estimated and calibrated out of the measurement system. The average timing delay between measurement signals was found experimentally using (38). With a constant velocity of 100 mm/s and an approximate constant offset error of 2.8 mm, the timing delay between measurement signals was estimated to be 28 ms.

From Figure 5 , it is shown that the laser tracker measurement is delayed behind the encoder measurement. Therefore, the encoder measured was delayed by 28 ms to in order to align the measurement signals. Once the measurements were aligned, the same constant velocity motion from the previous experiment was executed on the robot with L = 0.5, in an attempt to smooth over any small offsets still present in the measured kinematic error. The response of the KEO for this experiment is shown in Figure 8 . As observed in Figure 8 , delaying the encoder measurement reduced the original offsets in the z component's measured kinematic error. The percent reduction, between both experiments, of the average measured kinematic error was found to be 97.2% and 99.7% over the ranges of the first and second offsets ([7.85,18.3] and [22.1,32.31 ] seconds), respectively. Additionally, the KEO produced a smoother estimation of the kinematic error by rejecting 80.1% and 87.0% of the timing noise present in the measured kinematic error over the same ranges.

In this paper, a Kinematic Error Observer (KEO) was constructed using dynamic observer theory and implemented to effectively estimate the kinematic error of an industrial robot, even during robot motion. The observer's stability and performance were analyzed, and it was found that for positive gain the observer was stable and the 2% settling time was inversely proportional to the observer gain. In step experiments the difference in the theoretical and experimental 2% settling times was similar to the timing variations in the measurement devices. A timing analysis was performed on the mismatch in timing between the laser tracker measurements and robot encoder measurement systems. This analysis provided a method to remove the effect of the constant timing offset between these two devices. Experiments were conducted where the robot went through two motions. It was found that the KEO is capable of estimating the kinematic error during robot motion, and it effectively filters non-deterministic timing noise.

The error from timing mismatch was significantly reduced when calibrating for the constant component of the timing mismatch.

Author name / Procedia Manufacturing 00

COMET Project -Plug-and-produce COmponents and METhods for adaptive control of industrial robots enabling cost effective, high precision manufacturing in factories of the future

Hephaestus -EU H2020 Project

Boeing, Georgia Tech Unveil New Research Center

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The authors of this paper would like to thank Yaskawa/Motoman Robotics, Automated Precision Inc. (API), and the Center for Aerospace Manufacturing Technologies (CAMT) for providing equipment for the experimental system. The authors would also like to thank Missouri University of Science and Technology's Intelligent Systems Center for providing funding for this research.

Author name / Procedia Manufacturing 00 (2019) 000-000Inserting (24) into (30) and (31) , respectively, ( , ) ( ) ( , ) ( , )The terms containing o t in (32) and (33) can be combined into a single vector representing the kinematic error from the average timing delay,This term will offset the measured kinematic error by a constant value which is equivalent to the product of the spatial velocity and the average timing delay. Similarly, the terms containing t  in (32) and (33) can also be combined into a single vector representing the noise due to non-deterministic timing,This term will produce noise in the measured kinematic error that is proportional to the constant velocity. Combining (4), (34) , and (35), the time delayed measured kinematic error isThe axis-angle representation of a rotation matrix provides a more intuitive way to visualize and compare the orientation of a homogenous transformation [34] . Essentially, this representation describes any orientation by a single vector which defines a single rotation about an arbitrary axis in 3 . The elements of the resultant vector define the coordinates of the arbitrary axis while the vector's magnitude defines the rotation about this axis. Consider a generalized rotation matrix, 11 12 13 r r r r r r r r rThe single rotation about the arbitrary axis is calculated from the generalized rotation matrix by and arbitrary axis is calculated from the generalized rotation matrix and single rotation by 32 1 2sinx y z r r k r r k r r kTogether, (42) and (43) can be combined together into a single vector,which is the axis angle representation, r , of a generalized rotation matrix, R .