Abstract: Rehabilitation exercising equipment includes a main frame, a first geared member rotatably mounted on the main frame, a second geared member rotatably mounted on the main frame, a connecting mechanism mounted between the first geared member and the second geared member so that the first geared member and the second geared member are movable in concert with each other, a first idle geared member rotatably mounted on the main frame and connected with the connecting mechanism, a second idle geared member rotatably mounted on the main frame and connected with the connecting mechanism, and two handlebars secured on the first geared member and the second geared member respectively. Thus, a user's hands can hold the handlebars to pivot the handlebars in two opposite directions by connection of the connecting mechanism so as to achieve an exercising or rehabilitating function.

Abstract: An exercise apparatus to simulate climbing is described that includes such features as arm handles that move in synchronism with the motion of foot pedals to provide a total body workout; side handrails; a mounting step; linear foot movement at a simulated climbing angle; a three point support structure using a vertical support column; pedal track covers; a mechanism to provide constant resistance to pedal motion; and pedal impact absorption.

Abstract: A steering mechanism is provided that steers by imparting a variable amount of toe or camber to a wheel, and is capable of imparting both toe and camber using a single mechanism. Some versions of the steering mechanism can provide extremely high stability during turns controlled by the simple leaning of the operator. Vehicles using the mechanism are provided, including lean-steered vehicles, and including a three-wheeled mobile elliptically driven device.

Abstract: An exercise device includes an adjustable frame and an exercise equipment. The adjustable frame includes a main supporting frame, a handle frame and an adjustable seat assembly. The main supporting frame includes a frame member for facilitating a predetermined exercise type. The a handle frame is supported by the main supporting frame at a position above a first end portion of the frame member, and includes a first handle bar and a plurality of second handle bars. The adjustable seat assembly includes a seat frame having a coupling portion moveably coupled with the main supporting frame, and a seat member mounted on a support portion of the seat frame. By adjustably and movably coupling the seat frame with respect to the main supporting frame, a distance between the seat member and the handle frame can be freely adjusted.

Abstract: A multi-mode exercise device includes a main body having two supporting arm assemblies symmetrically mounted on two sides thereof. Each supporting arm assembly has four arms. The four arms of each supporting assembly are pivotally connected with each other via five pivot joints. The main body further has a guiding wheel set, a rotating disc and two connecting rods. Therefore, a user could step on pedals along vertical direction, parallel direction or circular direction selectively.

Abstract: In a first aspect, a method is provided that includes the steps of (1) providing a monitor for determining a level of exercise performance performed by an exerciser and for outputting a signal representative of the performance level; (2) using the monitor to measure a level of exercise performance performed by an exerciser; (3) outputting from the monitor a signal representative of the performance level; (4) providing a video game having a character, the video game inputting and being responsive to the performance level signal output by the monitor, the video game using the performance level signal to control a performance level of the video game character; and (5) while the exerciser is not exercising, controlling a performance level of the video game character based on a performance level signal output by the monitor.

Abstract: The invention pertains to a stationary exercise bike along with a display that provides instruction to lead a rider through an exercise program. The invention allows a rider to obtain benefits of a group, instructor-led class though the rider's schedule does not permit the rider to participate in the class. The invention also describes a method of exercising with the foregoing bike and display.

Abstract: A partially-inlaid elliptical gymnastics equipment system includes a plurality of pedals connected to moving arms that move in synchronism with said pedals, at least one fixed member configured within a reach of a user, a panel for controlling and monitoring the equipment system, a mechanical and electric assembly that connects at least one pair of pedals and moving arms to define an elliptical system and promotes an alternating displacement of the at least one pair of pedals and arms, a vertical partition that includes a first pair of vertical windows through which distal end parts of bars pass, each bar connecting one the pedals to the elliptical system and a pair of vertical windows through which the moving arms pass, the moving arms being synchronized with the movements of the pedals, and an aisle providing access to the mechanical and electric assembly.

Abstract: The disclosed invention is an exercise device being capable of providing aerobic exercise and abdominal muscle conditioning. The device is comprised of an upper frame and a lower frame. The upper frame includes a seat back which supports the upper portion of the torso of a user. The lower frame includes a pelvis support and a bicycle pedaling mechanism. The preferred embodiment includes a linkage system that connects the upper frame to the lower frame. This linkage is capable of providing a translating center of rotation when the upper frame rotates with respect to the lower frame. This is done to follow the body's translating center of rotation during trunk flexion. An alternative to the preferred embodiment includes a arcuate roller attached to the upper frame.

Abstract: An exercise workstation comprises a table assembly having a working surface, and an elliptical trainer. The elliptical trainer comprises a seat, and at least two footplates capable of elliptical motion about an axis having a horizontal component of motion greater than a vertical component of motion. A first distance between the seat and the table assembly is adjustable, a second distance between the seat and the axis is adjustable, the first distance and the second distance are independently adjustable, and the elliptical trainer is a recumbent or semi-recumbent elliptical trainer.

Abstract: An apparatus permitting a user to perform a simulated bicycling exercise is provided. The design includes a horizontal base, a lower front pivoting point, and user supporting hardware collectively adjoining pedals, a seat configured to support the user, and handlebars, the user supporting hardware provided above the horizontal base and interfacing with the lower front mounting point. The lower front pivoting point and the user supporting hardware include a multiple component resistive element arrangement and collectively define an axis of rotation forming an angle with the horizontal base of 30 to 45 degrees, sloping upward in a rearward direction from the front pivoting point to the user supporting hardware. When the user is seated on the seat and leans in one direction, such leaning causes the user supporting hardware, seat, and handlebars to rotate in the one direction about the axis of rotation.

Abstract: A saddle-positioning apparatus includes a first positioning unit for positioning a saddle in a first dimension, a second positioning unit for positioning the saddle in a second dimension, and a lever connected to both of the first and second positioning units. The lever is movable from a neutral position in a direction to move the first positioning unit from a locking position to a releasing position to allow movement of the saddle in the first dimension. The lever is movable from the neutral position in an opposite direction to move the second positioning unit from a locking position to a releasing position to allow movement of the saddle in the second dimension.

Abstract: The arm and leg-exercising machine is an apparatus that targets both the upper body and lower body to provide a full body workout. The upper and lower body workouts can be performed separately or simultaneously. The machine includes a generally U-shaped frame fixed to a base. Pedals and lateral handlebars are provided on the frame for respective leg (lower body) and arm (upper body) exercises. A three-drive mechanism including a plurality of pulleys or sprockets is coupled to the pedals and lateral handlebars and joined by bevel gears. A seat is mounted on the frame for strategically positioning an operator to employ the pedals and lateral handlebars.

Abstract: An apparatus permitting a user to perform a simulated bicycling exercise is provided. The design includes a frame having a head tube, a first upper rear mounting point and a second lower front mounting point configured to maintain the frame, a seat connected to said frame and configured to support the user in a forward facing orientation, a wheel positioned in association with said frame and pedals configured to interact with said wheel, and a stem connected to a handlebar arrangement, the stem passing through the headset collar and connected to a base by a connection arrangement comprising an arm. The frame is configured to pivot about the first upper rear mounting point and second lower front mounting point in response to leaning by the user, causing rotation of the stem within the head tube by movement of the connection arrangement comprising the arm.

Abstract: A rotatable handgrip for a cardiovascular exercise machine allowing a user to rotate the handgrip relative exercise arms of the machine about an axis which intersects the handle and is generally in line with their forearm.

Abstract: The present invention relates to a bike saddle structure in which the oscillation angle and height of a bike saddle are adjustable, and to an exercise bicycle comprising same. Vibrations in the bike saddle are minimized and the stability thereof is ensured during the left and right oscillation of the bike saddle, thus enabling a rider to exercise his/her whole body including the legs, pelvis, spine, abdomen, etc. over a long period of time. The forward and backward direction and height of the bike saddle are adjustable by the rider, thus providing a wide range of exercise.

Abstract: A bicycle trainer with a hill training apparatus is described. The hill training apparatus enables bicyclist to simulate uphill riding (incline), downhill riding (decline) and hill resistance or lack thereof. The hill training apparatus also provides for correct body positioning while riding uphill or downhill and with or without resistance. Riding on a trainer with correct body positioning provides for improved training.

Abstract: A vehicle simulation system equipped to display information on the rear side of a driver (rider). The simulation system includes a dummy bicycle operated by the rider, a monitor on the front side for displaying a scene based on the operations of the dummy bicycle, and a lamp unit connected to the dummy bicycle and provided on the rear side relative to the seated position of the rider. The lamp unit has a red lamp, a yellow lamp and a green lamp, and is controlledly turned ON under the action of a main control unit. An input device is provided on a steering handle of the dummy bicycle. The input device has a switch for red, a switch for yellow, and a switch for green. The rider checks behind, and operates the input device according to the ON/OFF conditions of the lamp unit.

Abstract: An exercise bike for rehabilitation includes a frame, on which a first exercise module designed for training hands, a second exercise module designed for training feet, and a damping device to provide the first and second exercise modules loadings are provided. The first and second exercise modules each has two opposite one-way clutches that user may use to independently train his/her hand or feet without interference.

Abstract: An optimal bicycle frame size based on operational characteristics provided by a rider when riding a dynamic fit unit is determined. A best-fit bicycle frame size that is a closest match to the optimal frame size is determined. At least one of the optimal X,Y location of the bicycle's handlebar relative to the bottom bracket, and the optimal X,Y location of the bicycle's seat relative to the bottom bracket, is determined. A stem and spacer from a list of available stems and spacers that will best fit between the head tube of the best-fit frame and the optimal X,Y location of the handlebar is determined. A seat post from a list of available seat posts that will best fit between the seat tube of the best-fit frame and the optimal X,Y location of the seat is determined. A list of the best-fit frame, the best fit stem and spacer, and the best fit seat post, is produced.

Abstract: An exercise bicycle is provided with a planar four bar linkage with cranks. The bicycle frame (5) is used as the machine frame of the planar four bar linkage, the seat (2) of the bicycle is fixed on a connecting rod (8) of the planar four bar linkage, and the connecting rod (8) is not directly connected with the bicycle frame (5). A middle axle (10) is used as the crank shaft, and one end of the connecting rod (8) is rotationally connected with a crank pin journal of the crank shaft (10), the other end thereof is connected with the member of the bicycle frame (5). The seat of the bicycle can do periodic movement with translation and rotation while a person rides the bicycle.

Abstract: An apparatus permitting a user to perform a simulated bicycling exercise is provided. The design includes a frame and a first mounting point and a second mounting point configured to maintain the frame. A seat is connected to the frame and configured to support the user. A wheel is positioned in association with said frame and pedals configured to interact with the wheel, and the frame is configured to pivot about the first mounting point and second mounting point in response to leaning by the user. Handlebars may be provided that enable further force application and enhance the leaning or pivoting in the bicycle riding simulation experience.

Abstract: A cycling exercise machine includes a housing rotatably attached to a shaft of a stationary base and having two channels, a carrier having two levers slidably engaged in the channels of the housing respectively, a foot support attached to the carrier, a latch attached to the housing, and a spring biasing member engaged with the latch for biasing and forcing the latch to engage with either of the levers and to anchor and retain the carrier and the levers and the foot support to the housing at the required or predetermined position and for allowing the users of different statuses or postures or sizes or dimensions or legs to suitably actuate or operate the cycling exercise machine.

Abstract: A supine cycling exercise machine includes a frame attached to a horizontal base and a rotation driving assembly supported by the frame. The driving assembly includes a rotatable crankshaft and two crankarms with proximal ends thereof attached to opposite ends of the crankshaft, with two pedals attached to distal ends of the crankarms. One or more crankarm includes a redirection connection mechanism at the proximal end thereof, and such a crankarm is adapted to be repositioned by an exerciser between an opposing and a parallel orientation relative to the other crankarm. The exercise machine can be used for two different cycling exercises, with either a reciprocating motion or a parallel motion of the exerciser's feet.

Abstract: An independent electric power type speed race apparatus is disclosed. The independent electric power type speed race apparatus comprises a first health cycle comprising a first pedal, a first cycle rotary body rotating by handling of the first pedal, and a first cycle load driving part working as a load checking the rotation of the first cycle rotary body; a first cycle speed sensor sensing a rotation speed of the first cycle rotary body; a second health cycle comprising a second pedal, a second cycle rotary body rotating by handling of the second pedal, and a second cycle load driving part working as a load checking the rotation of the second cycle rotary body; a second cycle speed sensor sensing a rotation speed of the second cycle rotary body; a display plate installed at the front of the first health cycle and the second health cycle.

Abstract: Provided is a novel physical therapy system and device that can be manually operated by a post-operative patient without the need of professional on site assistance, wherein the degree of movement of the limb or joint of interest is measured and recorded so as to provide a record of the progress of the prescribed post-operative therapy for immediate feedback to the patient as well as for long term data recordation for the therapist or physician.

Abstract: Embodiments of the invention may be used to both train and test instructor proficiency with a group exercise training routine. In a training mode, the graphical user interface may display visual cues indicating upcoming transitions from one activity to another. In response, the user may perform an action (e.g., clicking a button with a mouse cursor) to select the transition at the appropriate time. In a testing mode, the graphical user interface may display an indication of whether the correct selections are made, as well as made at the right time. This approach allows the user to rapidly develop substantial proficiency with the exercise routine.

Abstract: Disclosed is a motorized rehabilitation apparatus and method for disabled, impaired or injured individuals, which trains a proper gait, increases blood flow, relieves stress, and reconditions lower body muscles and joints. The device comprises a powered stationary bicycle having a seat, handle grips, and rotating foot pedals that receive motive input from an electric motor and user input. The device further includes a pair of thigh braces that are connected together between the user's thighs via a hingeable link and chain that controls and trains an individual's limbs through the pedal rotation. The disclosed method further combines the present bicycle device for rehabilitation in conjunction with visual stimuli in the way of a three dimensional television display that stimulates endorphins, relieves mental stress and allows the motive input from the bicycle and mild user input to exercise the limbs of a user without focusing on the rehabilitation activity.

Abstract: An exercising bike includes a main frame and a rear frame adjustably connected with the main frame. The main frame includes a support pipe having a slideway, a first linkage pivotally mounted on the support pipe, a seat mounted on the first linkage, a second linkage pivotally mounted on the support pipe, a handlebar mounted on the second linkage, and a third linkage pivotally connected with the first linkage and the second linkage. The rear frame includes an adjusting bar retractably mounted on the support pipe, a connecting lever pivotally connected with the adjusting bar and the first linkage, and at least one roller rotatably mounted on the adjusting bar and slidably mounted in the slideway. Thus, the position of the seat can be adjusted so that a user can operate the exercising bike in a sitting or lying posture.

Abstract: An elliptical exercise apparatus with a swinging effect has a pedals, a crank mechanism, a damping mechanism and pedals. A pair of sloping guide rails are arranged symmetrically between the front support and a middle support of a pedestal. One end of the pair of sloping guide rails is arranged laterally. A pair of pedals are arranged separately onto the pair of sloping guide rails and connected with the crank mechanism. A slide guide is steerably set at a front bottom of the pedals and slidably mated with the pair of sloping guide rails. A universal joint is set at the rear bottom of the pedals and is connected to the left and right cranks.

Abstract: A multifunctional elliptical trainer includes a frame, a seat, a leg training apparatus, and a hand training apparatus. The frame has at least roller rail. The seat is provided on the frame. The leg training apparatus includes a flywheel provided on the frame, two pedal bars, on each of which a pedal is provided, having ends pivotally connected to the flywheel, and having rollers to be engaged with the roller rail of the frame that the pedal bars slide along the roller rail. The hand training apparatus has two handles, each of which has a holding portion, pivoted on the frame and two linkages having opposite ends pivotally connected to the handles and the pedal bars respectively. User may sit on the seat, holds the handles, and steps on pedals of the pedal bars to train his/her legs and arms in an easy way.

Abstract: A bicycle handle assembly, of which the position of the handle grip can be freely adjusted according to the need of the rider, enabling the rider to ride the bicycle in a comfortable posture. The bicycle handle assembly includes: a stem prepared on a front-wheel fork of a bicycle; a handlebar installed on the stem; a handle grip coupled to either end of the handlebar; a ball joint prepared between the handlebar and the handle grip to enable the handle grip to rotate; and a securing element installed on the ball joint to secure the handle grip in a particular position. The bicycle handle assembly makes it possible to freely adjust the positions of the handle grips according to the need of the rider, thereby distributing the load that may otherwise be concentrated on the wrists and preventing pain and injury.

Abstract: The invention relates to a fitness device (1) that can be used in a mobile or stationary manner, comprising: a frame (4) on which two front wheels (6) and one rear wheel (5) are disposed, a drive device on each side of the frame (4), each made up of an arm drive rod (2) joined to a leg drive rod (3) by a hinge, wherein the leg drive rod (3) transfers a torque via a ring gear (10) and a chain to the rear wheel (5), a steering device (7) for steering the fitness device (1) in mobile use by means of tilting the frame (4), and a tilting/steering damper formed by damping elements (9) and disposed between the front wheels (6) and the frame (4).

Abstract: In an elliptical step exercise apparatus a dynamic link mechanism can be used to vary the stride length of the machine. A control system can also be used to vary stride length as a function of various exercise and operating parameters such as speed and direction as well as varying stride length as a part of a preprogrammed exercise routine such as a hill or interval training program. In addition the control system can use measurements of stride length to optimize operation of the apparatus.

Abstract: Various exercise machines have foot supporting linkages that move a person's feet through respective left and right paths of motion in respective planes that are skewed relative to one another.

Abstract: Described are several mechanisms for permitting a user to adjust the seat on a stationary exercise bicycle. The described mechanisms can be used to adjust the height of the seat or the fore and aft positioning of the seat on an upright type bicycle. Each of the described mechanisms can be configured to provide users with an optimum seat position and with a convenient latch mechanism to adjust the position of the seat.

Abstract: A cycle with tapered drivetrain connections. The cycle includes a frame, a crank rotatably connected to the frame, and a driven wheel rotatably connected to the frame. The crank includes a crank arm and a crank spindle rotatably connected to the frame and non-rotatably connected to the crank arm. The crank spindle includes a crank mating surface to interface with the crank arm. The diameter of the crank spindle continuously decreases along at least a portion of the crank mating surface in a lateral direction. The crank arm includes a crank bore to interface with the crank mating surface. At least a portion of the crank bore has a diameter that continuously decreases in lateral direction. At least a portion of the crank bore taper has a taper rate and diameter equal to a taper rate and diameter of at least a portion of the crank spindle taper.

Abstract: An exercise device includes a support frame configured to contact a support surface, and a cycler coupled to the frame. The cycler is configured to move with respect to the frame between a first position so as to be engaged by hands of a user of the exercise device, and a second position so as to be engaged by feet of the user. The exerciser also includes a lock configured to lock the cycler relative to the frame in the first and second positions. In an embodiment, the cycler is mounted on a movable frame that moves with respect to the support frame. In an embodiment, the movable frame is pivotally mounted to the support frame, such as to pivot generally parallel to the support frame, or generally across the support frame. In an embodiment, the movable frame is configured to slidably move up and down the support frame.

Abstract: An exercise device including at least one positionable component configured to be positioned by a user. The exercise device includes a frame to which a collar is mounted. The positionable component includes a member slidably received within the collar that may be positioned therein by sliding. A locking assembly is coupled to the collar and operable to lock the member in a selected position within the collar, release the member from the locked position, and when released, allow the member to slide within the collar. The locking assembly includes a cam pivotably mounted to the collar and a cam follower assembly selectively biased by the cam against a portion of the member disposed inside the collar. The locking assembly also includes a pair of engagement members disposed inside the collar opposite the cam. The engagement members are moveable relative to one another and biased by the cam against member.

Abstract: An elliptical trainer has a base, and two crank assemblies, a handle assembly and a linkage assembly mounted on the base. Each crank assembly has two cranks and a transmission wheel. A transmission strip is mounted around the transmission wheels of the crank assemblies to simulate rotations of the crank assemblies. The linkage assembly connects the cranks of the crank assemblies to the handle assembly. When the elliptical trainer is operated, the linkage assembly drives the cranks to alternately rotate upwardly and downwardly. The cranks that rotate downwardly drag the other cranks to rotate upwardly. Therefore, the crank assemblies and the linkage assembly operate smoothly, and consequently, the user is capable of smoothly performing walking or running movements.

Abstract: A modular sizing fit cycle, having an enclosed, acoustically dampened transmission, and calibrated height and angle adjusters, each designed to fit an individual to a bicycle according to specific measurements attained during a controlled ride. The sizing fit cycle is highly portable, having distinct pieces that facilitate portability. Additionally, the sizing fit cycle is preferably equipped with a quiet, chain and bell-driven transmission at the rear, facilitating accurate measurements via a load generator and providing a realistic simulation of a stable cycling session.

Abstract: A bicycle which is made up of a frame, a front and rear wheel, an electric DC motor, a multiple mode selection switch, a torque selection switch, and a rear wheel stand which can either raise the wheel off of a surface or allow the wheel to touch the surface, is described. The multi-purpose bicycle can be used in standard mode, electric mode, stationary gym mode, or mobile gym mode.

Abstract: A bicycle user information apparatus comprises an information receiver that receives information corresponding to the bicycle user through the Internet; and an administration control unit that manages the information received through the information receiver and enables external access to the managed information in response to predetermined criteria. The administration control unit includes a user registration unit that registers identifying information corresponding to the bicycle user.

Abstract: An elliptical exercise apparatus includes a frame unit having a base frame, two crank arms, and two link units disposed respectively on left and right sides of the frame unit and each including a pedal rod connected pivotally to one of the crank arms. An adjustment unit includes a motor mounted on the frame unit, a threaded rod connected drivenly to the motor, a pull unit connected pivotally to the frame unit and connected drivenly to the threaded rod, and two roller sets connected pivotally to and disposed respectively on left and right sides of the pull unit. Each roller set is disposed in rolling contact with the pedal rod of a respective link unit. The roller sets are limitedly movable toward or away from an axis of rotation of the crank arms by a pulling action of the pull unit.

Abstract: An exercise machine suitable for exercising a person's muscles contains a frame (100), a seat (102) situated over the frame, a seatback (104), a connection mechanism (106) for flexibly and adjustably connecting the seatback to the frame or/and the seat, and a pedaling mechanism (112) connectable to the frame. The pedaling mechanism has a pair of movable pedals (140) that revolve generally around a pedaling axis. The seatback is capable of swiveling.

Abstract: A cyclical skating motion exercise machine has a base frame assembly. A first pedal arm mounted on a first pedal axle, wherein the first pedal axle is substantially vertical. A first link assembly mounted to the first pedal arm at a first pedal joint. The first pedal joint provides motion between the first link assembly and the first pedal arm. A first pedal mounted to the first pedal arm for supporting a user's foot. A second pedal arm is mounted on a second pedal axle, wherein the second pedal axle is substantially vertical. A crank assembly has a first link assembly is mounted to the a crank assembly, and the crank assembly has rotational inertia is substantially orthogonally to the first pedal axle.

Abstract: A tent-like structure has a compact configuration and can be expanded to an expanded configuration large enough totally to enclose a patient. A recirculating gas supply arrangement supplies a mixture of oxygen and helium to the inside of the structure. The structure may contain exercise equipment, such as a static cycle, so that the patient can exercise in the gas mixture. The structure could also be used over a swimming pool. A weight or other means at the lower end of the structure helps to seal it with the floor surface and reduce escape of gas.

Abstract: An indoor cycling device including a unique frame arrangement with fore and aft adjustable seat and handlebar assemblies. The assemblies support a seat and handlebars for fore and aft movement. The assemblies may include a receiver with an elongate aperture with a slider positioned therein. The slider defines a first channel receiving a moveable member. A handle is operably coupled with the member to move the member within the channel in a first direction or a second direction such that a frictionally coupling is caused between the slider and the receiver when the slider is moved in the first direction and releases the coupling when the slider is moved in the second direction.

Abstract: The subject matter discloses an exercising device enabled to switch between elliptical course and circular course of footrests used by a user, comprising two main wheels, each connected to a different footrest; two connecting rods, the distal end of each connecting rod is connected to each main wheel and the proximal end of each connecting rod is connected to the body of the exercising device, said connecting rods are enabled to move the two main wheels on a horizontal axis.

Abstract: An apparatus permitting a user to perform a simulated bicycling exercise is provided. The design includes a horizontal base, a lower front pivoting point, and user supporting hardware collectively adjoining pedals, a seat configured to support the user, and handlebars, the user supporting hardware provided above the horizontal base and interfacing with the lower front mounting point. The lower front pivoting point and the user supporting hardware include a multiple component resistive element arrangement and collectively define an axis of rotation forming an angle with the horizontal base of 30 to 45 degrees, sloping upward in a rearward direction from the front pivoting point to the user supporting hardware. When the user is seated on the seat and leans in one direction, such leaning causes the user supporting hardware, seat, and handlebars to rotate in the one direction about the axis of rotation.