Model-free adaptive compound control method and system for position servo system
By employing a model-free adaptive composite control method, combining trajectory planning and interpolation with loop control, an observer and a composite controller are constructed. This addresses the issues of weak response and insufficient disturbance rejection in position servo systems, achieving both rapid response and strong disturbance rejection.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-30
- Publication Date
- 2026-03-13
AI Technical Summary
Existing position servo systems suffer from weak position response and low rigidity under traditional control methods, making it difficult to achieve zero overshoot, fast response, and strong anti-disturbance capabilities. Their performance is particularly insufficient in application scenarios such as vehicle-mounted LiDAR and assembly positioning.
A model-free adaptive composite control method is adopted. By integrating trajectory planning and interpolation, and loop control, a second-order extended state observer and composite controller are constructed. The observer bandwidth and controller bandwidth are adaptively adjusted to improve the dynamic response capability and disturbance rejection capability of the system.
It achieves zero overshoot, fast response and strong anti-disturbance capability of position servo system, improves the position response speed and stability of system and optimizes performance under different operating conditions.
Smart Images
Figure CN121657754A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of servo system control, and more specifically, relates to a model-free adaptive composite control method and system for a position servo system. Background Technology
[0002] Position servo systems are widely used in various fields of manufacturing and processing. Typically, the motion controller of the servo system generates position commands through trajectory planning and interpolation, while the servo driver performs position control, speed control, and current control. However, there is a problem of insufficient integration between the motion controller and the servo driver. In addition, in the traditional cascade three-loop control mode of position-speed-current, although precise tracking of position, speed, and current can be achieved, the bandwidth of the outer position loop is limited by the inner speed loop, resulting in low rigidity and weak position response capability of the servo system.
[0003] In point-to-point control applications such as vehicle-mounted LiDAR, assembly positioning, and automatic dispensing machines, the speed tracking performance of the position servo system is often not a primary concern; instead, the focus is on the system's position overshoot, position response speed, and anti-interference capability.
[0004] Therefore, for the above application scenarios, how to break through the limitations of traditional control methods and achieve control of the position servo system with no overshoot, fast position response speed, and strong anti-disturbance capability is an urgent problem to be solved. Summary of the Invention
[0005] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a model-free adaptive composite control method and system for position servo systems. Its purpose is to improve the performance of position servo systems from the perspectives of position overshoot, position response speed, and disturbance rejection capability by integrating controller trajectory planning and driver loop control and optimizing the loop structure of the servo system.
[0006] To achieve the above objectives, according to a first aspect of the present invention, a model-free adaptive composite control method for a position servo system is provided, comprising: S1, The motion controller obtains the position command, angular velocity command and angular acceleration command for each control cycle based on the target position and kinematic constraints of the position servo system using the specified trajectory planning method and online interpolation method; S2, the motion controller outputs the current control cycle. Position commands Angular velocity command and angular acceleration command ; S3, based on the previous control cycle k -1 rate of change of total disturbance observations and the current control cycle The observer bandwidth adaptive rate is used to calculate the current control cycle. Observer bandwidth ,according to Calculate the current control cycle Gain parameters of the observer , This completes the current control cycle. Construction of the observer; the previous control cycle k -1 filtered torque command value and actual angular velocity Input to the current control cycle The observer obtains the current control cycle. angular velocity observations and total system disturbance observations ; S4, based on the previous control cycle k -1 position tracking error and the current control cycle The composite controller's bandwidth adaptive rate calculation for the current control cycle The bandwidth of the composite controller ,according to Calculate the position control coefficient of the composite controller and angular velocity control coefficient This completes the current control cycle. The construction of a composite controller; the output of the motion controller , and Observer output and Input to the current control cycle The composite controller obtains the current control cycle. Unfiltered torque command value Filter it to obtain the current control cycle Filtered torque command value ; S5, the next control cycle k +1 is used as the current control cycle, returning to S2, until the target position of the position servo system is reached.
[0007] According to a second aspect of the present invention, a model-free adaptive composite control system for a position servo system is provided, comprising a computer program or instructions that, when executed by a processor, implement the method described in the first aspect.
[0008] According to a third aspect of the present invention, an electronic device is provided, comprising: a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is configured to read executable instructions stored in the computer-readable storage medium and execute the method as described in the first aspect.
[0009] According to a fourth aspect of the invention, a computer-readable storage medium is provided, the computer-readable storage medium storing computer instructions for causing a processor to perform the method as described in the first aspect.
[0010] In summary, compared with the prior art, the above-described technical solutions conceived by this invention can achieve the following beneficial effects: This invention provides a model-free adaptive composite control method for position servo systems, integrating trajectory planning and interpolation with loop control. Discrete position, angular velocity, and angular acceleration commands generated by trajectory planning and interpolation are sent to the composite controller. A second-order extended state observer is constructed to observe and estimate the angular velocity and total disturbance value of the servo system using the electromagnetic torque command value and the actual angular velocity, serving as state feedback input to the composite controller. An adaptive observer bandwidth rate is constructed using the variable rate of the observed total disturbance value. When the system is subjected to rapidly changing disturbances, the observer bandwidth is increased to improve the system's disturbance rejection capability. When the disturbance changes slowly, the observer bandwidth is decreased to improve the system's noise suppression capability. A composite controller based on a second-order hyperlocal model is constructed, using a position-velocity parallel control method combined with angular acceleration feedforward control and observer disturbance feedback compensation to improve the dynamic response capability of the position servo system and eliminate position overshoot. An adaptive controller bandwidth rate is constructed using the position tracking error. When the position tracking error is large during dynamic tracking, the controller bandwidth is increased to improve the system's dynamic response capability. When the tracking error is small after reaching the target position, the controller bandwidth is reduced to mitigate the impact of encoder noise during low-speed operation. Attached Figure Description
[0011] Figure 1 This is one of the flowcharts of the model-free adaptive composite control method for a position servo system provided in the embodiments of the present invention; Figure 2 This is the second flowchart of the model-free adaptive composite control method for a position servo system provided in this embodiment of the invention; Figure 3 (a) and (b) in the figure are control command signal curves obtained by trajectory planning and online interpolation when the motion controller adopts trapezoidal velocity planning and S-shaped velocity planning, respectively. Figure 4 (a) and (b) in the figure are open-loop Bode plots of the position servo system under different controller and observer bandwidths, respectively; Figure 5In the figure, (a), (b), and (c) are the command, disturbance, and noise response curves of the position servo system under different controller bandwidths, respectively. Figure 6 (a), (b), and (c) in the figure are the command, disturbance, and noise response curves of the position servo system under different observer bandwidths, respectively. Figure 7 (a), (b), and (c) in the figure are comparison figures of the position tracking experimental results of the position servo system under different control methods when using trapezoidal velocity planning. Figure 8 (a), (b), and (c) in the figure are comparative figures of position tracking experimental results of position servo systems under different control methods when sampling S-shaped velocity planning; Figure 9 (a) and (b) in the figure are comparison figures of the disturbance suppression experimental results of the position servo system under different control methods. Detailed Implementation
[0012] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0013] This invention provides a model-free adaptive composite control method for a position servo system, such as... Figure 1 As shown, it includes: S1, the motion controller obtains the position command, angular velocity command and angular acceleration command for each control cycle based on the target position and kinematic constraints of the position servo system using the specified trajectory planning method and online interpolation method.
[0014] Specifically, in step S1, the motion controller obtains discrete command signals, namely position commands, angular velocity commands, and angular acceleration commands for each control cycle, based on the target position and kinematic constraints of the position servo system using a specified trajectory planning method and an online interpolation method.
[0015] The specified trajectory planning method can be any existing trajectory planning method, such as trapezoidal velocity planning, S-shaped velocity planning, or quintic polynomial planning.
[0016] For example, when the trajectory planning method employs trapezoidal velocity planning and S-shaped velocity planning respectively, the position, angular velocity, and angular acceleration command signals obtained through trajectory planning and online interpolation are as follows: Figure 3 (a) and Figure 3 As shown in (b) of the diagram.
[0017] S2, the motion controller outputs the current control cycle. Position commands Angular velocity command and angular acceleration command .
[0018] S3, based on the previous control cycle k -1 rate of change of total disturbance observations and the current control cycle The observer bandwidth adaptive rate is used to calculate the current control cycle. Observer bandwidth ,according to Calculate the current control cycle Gain parameters of the observer , This completes the current control cycle. Construction of the observer; the previous control cycle k -1 filtered torque command value and actual angular velocity Input to the current control cycle The observer obtains the current control cycle. angular velocity observations and total system disturbance observations .
[0019] Specifically, in step S3, the current control cycle is constructed based on the mechanical motion equations of the permanent magnet motor. An extended state observer is used to obtain the current control cycle. angular velocity observations and total system disturbance observations .
[0020] The mechanical motion equations of a permanent magnet motor are:
[0021] in, J The moment of inertia of the servo system; This represents the actual rate of change of angular velocity. This refers to the actual electromagnetic torque of the servo system. This refers to the disturbance torque of the servo system.
[0022] The current control cycle being constructed The observer is: ; in, To control the gain, This is an estimate of the rotational inertia of the servo system. For the previous control cycle k The observed angular velocity value of -1 For the previous control cycle k -1 actual angular velocity, For the previous control cycle k The angular velocity observation error value is -1. , All are current control cycles The gain parameters of the observer.
[0023] Current control cycle Gain parameters of the observer , The bandwidth method is used for tuning, and the calculation formula is as follows:
[0024] in, For the current control cycle The observer bandwidth, based on the current control cycle The observer bandwidth adaptive rate is calculated.
[0025] Current control cycle The observer bandwidth adaptation rate is: ; in, This represents the minimum bandwidth of the observer. This represents the maximum bandwidth of the observer. The observer bandwidth adaptive rate gain coefficient, To control the gain, From difference equation ,in , To connect the rate of change of the total disturbance observation The cutoff frequency of the subsequent first-order low-pass filter, The duration of a control cycle, initial value The initial value of the observer bandwidth is set manually.
[0026] Indicates the previous control cycle k The rate of change of the observed total disturbance of the system is susceptible to encoder sampling noise, therefore low-pass filtering is performed first. When the current period is large, the system is subjected to rapidly changing disturbances, requiring an increase in the current period. k The observer bandwidth is used to suppress the effects of disturbances when When the current period is small, the current period can be appropriately reduced. kThe observer bandwidth is increased to mitigate the impact of factors such as encoder sampling noise. Based on the aforementioned current cycle... k The observer bandwidth adaptive rate is calculated to obtain the current cycle. k The observer bandwidth is calculated, and the current period is recalculated using the bandwidth method. k Gain parameters of the observer , .
[0027] In summary, in step S3, based on the current control cycle... The observer bandwidth adaptive rate and the previous control cycle k The rate of change of the total disturbance observations is -1, which is used to calculate the current control period. Observer bandwidth Then, based on the usage Bandwidth method for calculating the current control cycle Gain parameters of the observer , Complete the current control cycle The observer is constructed; then, the previous control cycle is... k -1 filtered torque command value Input to the current control cycle The observer obtains the current control cycle. angular velocity observations and total system disturbance observations .
[0028] S4, based on the previous control cycle k -1 position tracking error And the composite controller bandwidth adaptive rate calculation for the current control cycle The bandwidth of the composite controller ,according to Calculate the position control coefficient of the composite controller and angular velocity control coefficient This completes the current control cycle. The construction of a composite controller; , , , and Common input to the current control cycle The composite controller obtains the current control cycle. Unfiltered torque command value Filter it to obtain the current control cycle Filtered torque command value .
[0029] Specifically, in step S4, the current control cycle is constructed. A composite controller based on a second-order hyperlocal model: ; in, For the current control cycle The encoder position feedback value, The angular acceleration feedforward coefficient is... ; , These are the position control coefficient and angular velocity control coefficient of the composite controller, respectively.
[0030] Position control coefficient and angular velocity control coefficient of the composite controller , The tuning method is as follows:
[0031] in, The damping coefficient of the composite controller has a range of values. ; For the current control cycle The bandwidth of the composite controller is determined based on the previous control cycle. k -1 position tracking error and the current control cycle The bandwidth adaptive rate of the composite controller is calculated.
[0032] Current control cycle The bandwidth adaptation rate of the composite controller is: ; in, This represents the minimum bandwidth of the composite controller. This represents the maximum bandwidth of the composite controller. This is the controller bandwidth adaptive rate gain coefficient; the initial value of the controller bandwidth is set manually.
[0033] When the system accelerates or decelerates or is subjected to external disturbances, the system's position tracking error... If the increase is sudden, the current control period should be appropriately increased. The bandwidth of the composite controller can enhance the system's dynamic response and disturbance rejection capabilities. When the system operates at low speed or zero speed, the signal-to-noise ratio of the encoder feedback signal is low; in this case, the control cycle should be appropriately reduced. The bandwidth of the composite controller is increased to avoid excitation encoder noise. Based on the current control cycle described above... The composite controller's bandwidth adaptive rate and the previous control cycle k -1 position tracking error The current control cycle can be calculated. The bandwidth of the composite controller is determined, and the current control cycle is recalculated using the aforementioned control parameter tuning method. Control parameters of the composite controller , .
[0034] In summary, in step S4, based on the previous control cycle... k -1 position tracking error and the current control cycle The composite controller's bandwidth adaptive rate calculation for the current control cycle Composite controller bandwidth Then use And the method for calculating the current control cycle using the composite controller parameter tuning method Position control coefficient of the composite controller Angular velocity control coefficient This completes the current control cycle. The composite controller is constructed; then, the current control cycle output by the motion controller in step S2 is used. Position commands Angular velocity command and angular acceleration command And the angular velocity observation value of the position servo system output by the observer in step S3. Total disturbance observations Input to the current control cycle The composite controller obtains the current control cycle. Unfiltered torque command value Filter it to obtain the current control cycle Filtered torque command value The current loop calculation and modulation module in the position servo system is based on... The inverter bridge switching sequence is calculated to achieve vector control.
[0035] S5, the next control cycle k +1 is used as the current control cycle, returning to S2, until the target position of the position servo system is reached.
[0036] The method provided by this invention assumes that the actual electromagnetic torque of the motor can accurately track the torque command value. Therefore, the angular velocity estimation equation and disturbance estimation equation of the extended state observer are: ; The method provided by this invention, when considering torque command filtering, The larger the value, the greater the system crossover frequency and phase margin, and the greater the closed-loop bandwidth of the system. The larger the value, the higher the system crossover frequency, and the slightly lower the system phase margin. The open-loop Bode diagrams of the lower system are as follows: Figure 4 (a) and Figure 4 As shown in (b) of the diagram. The phase margin of the system is at its maximum at that time, and to avoid position overshoot, it will be... Set to 1.
[0037] Under the model-free adaptive composite control method for position servo systems provided by this invention, the system's position response expression is:
[0038] in, G r ( s ), G d ( s )and G n ( s These are the system's command response transfer function, disturbance response transfer function, and noise response transfer function, respectively.
[0039] in, γ Inertia gain ratio ; T f The torque command filtering time constant; Let be the characteristic polynomial of the system, and .
[0040] Both the controller and the observer use the above parameter tuning method to maintain... ω o = 800π rad / s remains constant. ω c = 60π rad / s ω c = 80π rad / s ω c = 100π rad / s, the command response, disturbance response, and noise response curves of the system are as follows Figure 5 As shown in (a) to (c), it can be seen that ω c Larger systems generally have stronger position tracking and disturbance suppression capabilities, but weaker noise suppression capabilities. It is particularly important to note that... ω c When the value is large, the system command response curve exhibits a significant peak, which may lead to overshoot in the system's position response. Similarly, maintaining... ωc = 80π rad / s remains constant. ω o = 600π rad / s ω o = 800π rad / s ω o The command response, disturbance response, and noise response curves of the system at 1000π rad / s are shown below. Figure 6 As shown in (a) to (c), it can be seen that ω o The larger the system, the stronger its disturbance suppression capability and the weaker its noise suppression capability. The observer bandwidth does not affect the system's command response. From the above analysis, it can be seen that there is a coupling relationship between the system's command response, disturbance response, and noise response. Under different system operating states, the adaptive adjustment of the controller and observer bandwidth can achieve a balance and optimization among various performance requirements.
[0041] To verify the effectiveness and superiority of the control method proposed in this invention, the following experiments were conducted: First, a position tracking experiment was conducted under trapezoidal velocity programming to verify the effectiveness of the novel control loop structure and controller bandwidth adaptation. The target position is... θ r = 10π rad, angular velocity limit is ω max = 60π rad / s, angular acceleration limit is a max = 400π rad / s 2 The observer bandwidth is ω o = 800π rad / s, torque filtering time is T f = 0.5ms, angular acceleration feedforward coefficient is k a = 1.0, respectively in the controller bandwidth ω c = 80π rad / s, controller bandwidth ω c = 120πrad / s, with controller bandwidth adaptive rate and parameters set to ω cmax = 160π rad / s ω cmin = 80π rad / s C c The experiment was conducted at 160°C, and the results are as follows: Figure 7 As shown in (a) to (c) of the text.
[0042] Subsequently, a position tracking experiment was conducted under S-shaped velocity planning to verify the effectiveness of the novel control loop structure and controller bandwidth adaptation. The target position was... θ r = 10π rad, angular velocity limit is ω max = 60π rad / s, angular acceleration limit is a max = 400π rad / s 2 Angular acceleration is limited to j max = 4000π rad / s 3 The observer bandwidth is ω o = 800πrad / s, torque filtering time is T f = 0.5ms, angular acceleration feedforward coefficient is k a = 1.0, respectively in the controller bandwidth ω c = 80π rad / s, controller bandwidth ω c = 120π rad / s, add controller bandwidth adaptive rate and set the parameter to ω cmax = 160π rad / s ω cmin = 80π rad / s C c The experiment was conducted at 160°C, and the results are as follows: Figure 8 As shown in (a) to (c) in the figure.
[0043] Depend on Figure 7 and Figure 8 Experimental results show that, under suitable parameters, the position response of the system under the new control loop architecture does not exhibit overshoot. Furthermore, after incorporating the controller bandwidth adaptive rate, the maximum position tracking error of the system under trapezoidal velocity programming is reduced by […]. ω c = 2.72 × 10 at 80π rad / s -3 rad, ω c = 2.54 × 10 at 120π rad / s -3 Rad decreased to 1.54 × 10 -3 The maximum position tracking error of the system under S-shaped velocity planning is given by rad and rad respectively. ω c = 1.83 × 10 at 80π rad / s -3 rad, ω c = 1.69 × 10 at 120π rad / s -3 Rad decreased to 9.74 × 10 -4 rad, and did not appear ω c At a speed of 120π rad / s, the large controller bandwidth causes oscillations due to encoder noise at zero speed. The proposed control method improves the upper limit of the controller's bandwidth, resulting in a system with no overshoot and a fast response speed.
[0044] Finally, disturbance suppression experiments were conducted to verify the effectiveness of the novel control loop structure and the adaptive observer bandwidth. The controller bandwidth was... ω c = 110π rad / s, angular acceleration feedforward coefficient is k a = 1.0, torque filtering time is T f =0.5ms, respectively within the observer bandwidth ω o = 800π rad / s, add observer bandwidth adaptive rate and set parameters as ω omax =1400π rad / s ω omin = 600π rad / s, Co = 0.01, ω ocut The experiment was conducted at 400π rad / s. After the motor reached the target position, an external load torque of 0.3 N·m was suddenly applied. The experimental results are as follows: Figure 9 (a) Figure 9 As shown in (b) of the diagram.
[0045] Depend on Figure 9 Experimental results show that the system under the new control loop architecture has strong anti-disturbance capability, and after adding the observer bandwidth adaptive rate, the maximum position error of the system after being subjected to external disturbances is reduced from... ω o = 3.16 × 10 at 800π rad / s -3 Rad decreased to 2.56 × 10 -3 The proposed control method achieves higher bandwidth (rad) and eliminates the oscillations caused by encoder noise at zero speed due to the large observer bandwidth. Furthermore, the upper limit of the observer bandwidth is increased, resulting in stronger system noise immunity.
[0046] The above experimental results show that, by using the method provided by the present invention, the upper limit of the bandwidth of the system's controller and observer is significantly improved, the system's position response has no overshoot, the position response speed is fast, and the anti-disturbance capability is strong.
[0047] This invention provides a model-free adaptive composite control system for a position servo system, including a computer program or instructions, which, when executed by a processor, implement the method described in any of the above embodiments.
[0048] This invention provides an electronic device, including: a computer-readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is configured to read executable instructions stored in the computer-readable storage medium and execute the method as described in any of the above embodiments.
[0049] This invention provides a computer-readable storage medium storing computer instructions that cause a processor to perform the method described in any of the above embodiments.
[0050] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A model-free adaptive composite control method for a position servo system, characterized in that, include: S1, The motion controller obtains the position command, angular velocity command and angular acceleration command for each control cycle based on the target position and kinematic constraints of the position servo system using the specified trajectory planning method and online interpolation method; S2, the motion controller outputs the current control cycle. Position commands Angular velocity command and angular acceleration command ; S3, based on the previous control cycle k -1 rate of change of total disturbance observations and the current control cycle The observer bandwidth adaptive rate is used to calculate the current control cycle. Observer bandwidth ,according to Calculate the current control cycle Gain parameters of the observer , This completes the current control cycle. Construction of the observer; the previous control cycle k -1 filtered torque command value and actual angular velocity Input to the current control cycle The observer obtains the current control cycle. angular velocity observations and total system disturbance observations ; S4, based on the previous control cycle k -1 position tracking error and the current control cycle The composite controller's bandwidth adaptive rate calculation for the current control cycle The bandwidth of the composite controller ,according to Calculate the position control coefficient of the composite controller and angular velocity control coefficient This completes the current control cycle. The construction of a composite controller; the output of the motion controller , and Observer output and Input to the current control cycle The composite controller obtains the current control cycle. Unfiltered torque command value Filter it to obtain the current control cycle Filtered torque command value ; S5, the next control cycle k +1 is used as the current control cycle, returning to S2, until the target position of the position servo system is reached.
2. The method as described in claim 1, characterized in that, The current control cycle The observer bandwidth adaptation rate is: ； in, This represents the minimum bandwidth of the observer. This represents the maximum bandwidth of the observer. The observer bandwidth adaptive rate gain coefficient, To control the gain, From difference equation ,in , To continue from the previous control cycle k -1 rate of change of total disturbance observations The cutoff frequency of the subsequent first-order low-pass filter, The duration of one control cycle; Current control cycle Gain parameters of the observer , The calculation formula is: 。 3. The method as described in claim 2, characterized in that, The current control cycle The observer is: ； in, For the previous control cycle k The observed angular velocity value of -1 For the previous control cycle k -1 actual angular velocity, For the previous control cycle k -1 is the angular velocity observation error value.
4. The method as described in claim 1, characterized in that, The current control cycle The bandwidth adaptation rate of the composite controller is: ； in, This represents the minimum bandwidth of the composite controller. This represents the maximum bandwidth of the composite controller. This refers to the controller bandwidth adaptive rate gain coefficient. The current control cycle Position control coefficient of the composite controller and angular velocity control coefficient The calculation formula is: ； in, is the damping coefficient of the composite controller.
5. The method as described in claim 4, characterized in that, Current control cycle The composite controller is: ； in, For the current control cycle The encoder position feedback value, is the angular acceleration feedforward coefficient.
6. A model-free adaptive composite control system for a position servo system, comprising a computer program or instructions, characterized in that, When the computer program or instructions are executed by a processor, they implement the method as described in any one of claims 1-5.
7. An electronic device, characterized in that, include: Computer-readable storage media and processors; The computer-readable storage medium is used to store executable instructions; The processor is configured to read executable instructions stored in the computer-readable storage medium and execute the method as described in any one of claims 1-5.
8. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer instructions for causing a processor to perform the method as described in any one of claims 1-5.